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(of)h(dif)n(fer)n(ent)g(biases)f(for)h(all)h(the)e(coins)h(to)f(be)300 1577 y(\002nite)h(\226)h(one)e(can)i(think)f(of)g(this)g(r)n (estriction)g(as)g(having)h(only)f(a)h(\002nite)f(number)g(of)g(mints,) 300 1746 y(each)e(pr)n(oducing)f(coins)h(with)f(a)i(dif)n(fer)n(ent)e (bias.)47 b(Since)29 b(all)h(coins)f(pr)n(oduced)e(by)h(the)g(same)300 1916 y(mint)d(ar)n(e)g(identically)h(biased,)f(we)f(can)i(r)n(efer)e (to)g(them)g(collectively)i(as)f FA(\()p Fz(q)2939 1930 y Fr(i)2967 1916 y Fz(;)15 b(p)3053 1930 y Fr(i)3081 1916 y FA(\))p FF(;)26 b(and)f(hence,)300 2085 y(we)j(can)i(let)f FD(D)39 b FA(=)e FD(f)p FA(\()p Fz(q)1083 2099 y Fr(i)1111 2085 y Fz(;)15 b(p)1197 2099 y Fr(i)1225 2085 y FA(\))p FD(g)30 b FF(be)f(the)f(set)g(of)h(all)h(possible)e(coin)h(biases.)47 b(A)28 b(particle)i(moving)300 2254 y(accor)n(ding)j(to)f(these)f(r)o (ules)h(is)h(an)g(example)f(of)h(a)g(random)f(walk)h(in)g(a)h (nonhomogeneous)300 2424 y(envir)n(onment)22 b(\(with)h(a)g(single)f (internal)h(state\).)583 2593 y(Another)30 b(extension)e(to)i(the)g (random)h(walk)g(involves)g(giving)g(the)f(particle)h(an)g(in-)300 2762 y(ternal)f(state)f Fz(s)38 b FD(2)h(f)p FA(1)p Fz(;)15 b(:)g(:)g(:)i(;)e(m)p FD(g)p FF(.)51 b(Now)-8 b(,)30 b(not)f(only)h(will)h(the)f(particle)g(move)g(back)h(and)f(forth)300 2932 y(on)c(the)f(integer)g(line,)i(but)g(it)f(will)h(also)g(change)f (its)g(internal)g(state.)38 b(The)25 b(r)n(ole)i(of)f(the)f(coin)i(is) 300 3101 y(expanded)h(to)i(dictate)g(the)f(movement)h(between)f(the)g (internal)i(states)d(as)j(well)f(as)h(the)e(left)300 3271 y(and)g(right)g(movement)g(of)g(the)g(particle.)48 b(In)29 b(this)g(scenario,)h Fz(\013)38 b FA(=)g(\()p Fz(q)2764 3285 y Fr(ij)2824 3271 y FA(\))30 b FF(and)f Fz(\015)43 b FA(=)37 b(\()p Fz(p)3351 3285 y Fr(ij)3412 3271 y FA(\))30 b FF(ar)n(e)300 3440 y(non-negative)22 b(matrices)i(wher)n(e)e Fz(q)1530 3454 y Fr(ij)1614 3440 y FF(is)h(the)f(pr)n(obability)j(of)e(the)f(particle)i(moving)f(into)g (state)300 3609 y Fz(j)36 b FF(fr)n(om)31 b(state)f Fz(i)h FF(and)f(step)f(towar)n(ds)g(the)h(left,)i(and)f Fz(p)2191 3623 y Fr(ij)2282 3609 y FF(is)f(the)g(corr)n(esponding)e(right-move) 300 3779 y(pr)n(obability)-10 b(.)39 b(In)25 b(addition,)h(the)f(sum)h (of)g(these)e(two)h(matrices)h(will)i(be)d(a)i(stochastic)e Fz(m)e FD(\002)f Fz(m)300 3948 y FF(matrix.)31 b(W)-8 b(e)23 b(now)g(have)h(a)g(general)f(random)g(walk)h(in)g(a)g (nonhomogeneous)c(envir)n(onment,)300 4117 y(with)j(a)g(\002nite)f (internal)h(state)f(space.)583 4287 y(Finally)-10 b(,)22 b(suppose)c(we)i(have)h(in\002nitely)g(many)g(particles,)g(each)g (involved)g(in)g(its)g(own)300 4456 y(random)j(walk)g(on)f(a)h(dif)n (fer)n(ent)g(nonhomogeneous)c(envir)n(onment,)k(but)f(with)h(the)f (same)g(set)300 4625 y FD(D)s FF(.)53 b(This)31 b(collection)g(of)g (random)g(walks)g(is)g(r)n(eferr)n(ed)f(to)h(as)g(a)g(\223random)h (set-walk\224)e(in)i(the)300 4795 y(theory)21 b(developed)f(in)k(the)d (subsequent)g(chapters,)g(and)i(is)f(the)g(main)i(topic)f(of)f(this)g (thesis.)583 4964 y(If)31 b(we)f(ignor)n(e)g(left-right)g(movement)g (of)g(all)i(the)e(particles,)i(the)d(random)i(set-walk)1927 5349 y(2)p eop %%Page: 3 8 3 7 bop 300 391 a FF(r)n(educes)24 b(to)g(the)h(Markov)g(set-chain)g (as)g(intr)n(oduced)f(by)h(D.)h(Hart\002el)e([22)r(],)h(with)h(the)e (transi-)300 561 y(tion)f(set)f FA(\006)j(=)h FD(f)p Fz(\013)c FA(+)e Fz(\015)28 b FD(j)c FA(\()p Fz(\013;)15 b(\015)5 b FA(\))27 b FD(2)f(D)s(g)p FF(.)j(On)24 b(the)e(other)g (hand,)h(if)h(the)e(set)g Fz(D)k FF(has)d(only)f(one)h(coin)300 730 y(\226)h(that)f(is,)h(if)g(spatial)g(nonhomogeneity)d(is)i(taken)g (out,)g(the)f(system)g(is)h(known)g(as)g(a)h(random)300 899 y(walk)29 b(with)f(internal)h(states,)f(as)h(studied)e(by)h(D.)h (Sz)8 b(\264)-38 b(asz)29 b(and)g(A.)f(Kr)8 b(\264)-38 b(amli)30 b([28)q(,)e(29)q(,)h(27)q(].)45 b(This)300 1069 y(lineage)23 b(is)g(r)n(e\003ected)e(in)i(our)g(choice)g(of)g(the) e(term)i(\223random)g(set-walk\224)f(as)h(the)f(name)h(of)g(the)300 1238 y(class)g(of)g(random)f(pr)n(ocesses)f(of)h(inter)n(est.)300 1481 y Fp(Remark.)41 b FF(While)22 b(we)g(have)h(been)e(using)h(coins)g (with)g(two)f(possible)h(outcomes)f(\(with)h(pr)n(ob-)300 1651 y(abilities)g Fz(\013)g FF(and)e Fz(\015)5 b FF(\))22 b(in)f(the)f(pr)n(evious)g(examples,)g(the)g(actual)j(theory)c(uses)g (the)h(mor)n(e)h(general)300 1820 y(notion)26 b(of)h(a)h (\223transition)f(triple\224)g FB(T)33 b FA(=)g(\()p Fz(\013;)15 b(\014)5 b(;)15 b(\015)5 b FA(\))p FF(,)30 b(wher)n(e)c Fz(\013)h FF(and)g Fz(\015)32 b FF(ar)n(e)c(as)f(befor)n (e,)g(and)g Fz(\014)32 b FF(is)300 1990 y(the)f(pr)n(obability)i(\(or)e (the)g(matrix)i(of)e(pr)n(obabilities\))i(that)f(the)f(particle)h (stays)e(at)i(the)f(same)300 2159 y(position.)40 b(The)26 b(analogue)h(of)g(the)f(set)g FD(D)k FF(described)c(above)h(is)g(the)f (\223transition)h(set\224)g FD(T)22 b FF(.)41 b(All)300 2328 y(these)21 b(terms)h(will)h(be)g(de\002ned)e(rigor)n(ously)g(in)i (Chapter)g(2.)300 2702 y Ft(1.2)119 b(Motivation)300 2969 y FF(The)23 b(development)f(of)i(the)f(theory)f(of)h(random)h (set-walk)f(was)h(motivated)f(by)h(the)f(analysis)300 3138 y(of)k(bounded-err)n(or)d(two-way)i(nondeterministic)g(pr)n (obabilistic)j(\002nite)d(automata)h(\(2npfa\),)300 3308 y(as)j(studied)f(by)h(Condon)f(et)g(al.)52 b([10)q(].)e(A)30 b(2npfa)h(is)f(a)h(\002nite)f(automata)g(with)h(both)e(nonde-)300 3477 y(terministic)g(and)h(pr)n(obabilistic)h(states,)e(and)g(a)g (two-way)g(head.)47 b(When)28 b(the)h(analysis)g(was)300 3646 y(focused)18 b(on)h(the)f(unary)h(languages)f(accepted)h(by)g(a)g (bounded-err)n(or)f(2npfa,)i(it)f(was)g(r)n(ealized)300 3816 y(that)27 b(the)f(operation)h(of)g(a)h(2npfa)g(exhibits)f(a)g (random)g(walk)h(behaviour)g(when)e(its)h(input)g(is)300 3985 y(a)d(string)e FA(0)671 3952 y Fr(n)742 3985 y FF(fr)n(om)i(the)f (alphabet)h FD(f)p FA(0)p FD(g)p FF(.)31 b(This)23 b(behaviour)h(is)g (captur)n(ed)f(by)g(the)f(theory)g(of)h(ran-)300 4154 y(dom)29 b(set-walk,)i(wher)n(e)e(the)g(walk)h(component)e(corr)n (esponds)f(to)i(the)g(head)g(moving)h(back)300 4324 y(and)d(forth)g(on) f(the)h(input)f(tape,)i(and)f(wher)n(e)f(the)g(internal)i(state)e (component)f(corr)n(esponds)300 4493 y(the)d(state)f(of)i(the)f (automaton.)583 4663 y(It)g(is)g(conjectur)n(ed)f(that)g(all)j(unary)d (languages)h(accepted)f(by)h(a)g(2npfa)h(ar)n(e)f(r)n(egular)-7 b(.)28 b(A)300 4832 y(key)21 b(step)g(in)j(pr)n(oving)e(this)g (conjectur)n(e)g(is)h(to)f(establish)h(the)f(validity)h(of)g(the)f (following:)527 5100 y Fw(The)29 b(sets)h(of)e(\002rst-exit)j(pr)n (obability)h(matrices)e(corr)n(esponding)i(to)d(a)f(sequence)h(of)g (ran-)1927 5349 y FF(3)p eop %%Page: 4 9 4 8 bop 527 391 a Fw(dom)29 b(set-walks)i(of)e(lengths)h Fz(d;)15 b FA(2)p Fz(d;)g FA(3)p Fz(d;)g(:)g(:)g(:)32 b Fw(conver)n(ges)f(in)e(the)g(Hausdorff)h(metric,)h(for)527 561 y(some)24 b(periodicity)i Fz(d)p Fw(.)583 804 y FF(A)g(special)h (case)f(of)g(this)g(r)n(esult,)g(for)g(the)g(case)g(when)f(the)g (internal)i(state)e(space)h(con-)300 974 y(tains)d(only)f(one)g(state,) f(is)i(pr)n(oved)f(in)h(Chapter)f(3.)300 1343 y Ft(1.3)119 b(Related)30 b(W)-11 b(ork)300 1615 y Fo(1.3.1)99 b(Markov)25 b(Set-Chains)300 1848 y FF(The)h(theory)f(of)h(Markov)h(set-chains)f (has)h(been)f(studied)f(for)h(over)g(twenty)f(years)h(by)g(Hart-)300 2018 y(\002el)35 b([22)q(])g(and)f(others.)62 b(A)34 b(Markov)g(set-chain)h(is)f(a)h(sequence)e FA(\()p Fz(M)2739 1985 y Fr(i)2768 2018 y FA(\))2803 1985 y Fn(1)2803 2043 y Fr(i)p Fs(=0)2956 2018 y FF(of)h(sets)f(of)i(\002nite,)300 2187 y(substochastic)h(matrices,)k(wher)n(e)c Fz(M)1660 2154 y Fr(k)r Fs(+1)1844 2187 y FA(=)52 b Fz(M)10 b(M)2163 2154 y Fr(k)2257 2187 y FA(=)51 b FD(f)p Fz(A)2492 2201 y Fs(1)2532 2187 y Fz(A)2600 2201 y Fs(2)2676 2187 y FD(j)37 b Fz(A)2806 2201 y Fs(1)2897 2187 y FD(2)51 b Fz(M)5 b(;)15 b(A)3210 2201 y Fs(2)3302 2187 y FD(2)51 b Fz(M)3512 2154 y Fr(k)3555 2187 y FD(g)300 2356 y FF(is)34 b(the)g(set)f(of)h(all)i(possible)d(pr)n(oducts)g(of)h(matrices)g(in)h (the)f(set)f Fz(M)44 b FF(of)34 b(length)g Fz(k)s FF(.)62 b(Hart\002el)300 2526 y(pr)n(oved)31 b(many)i(r)n(esults)e(concerning)g (Markov)h(set-chains,)i(including)f(ones)e(on)h(the)f(crite-)300 2695 y(ria)39 b(for)f(guaranteeing)e(conver)n(gence)h(of)h(a)h(Markov)e (set-chain,)42 b(and)c(decomposition)e(of)300 2864 y(the)30 b(state)f(space)h(into)g(er)n(godic)g(and)g(transient)g(classes.)50 b(In)30 b(particular)-7 b(,)34 b(he)c(pr)n(oved)f(that)i(a)300 3034 y(Markov)h(set-chain)g(is)g(conver)n(gent)f(if)i(it)f(is)h(pr)n (oduct-scrambling,)h(i.e.)56 b(if)33 b(ther)n(e)e(exists)g(an)300 3203 y Fz(r)k FD(2)d Fm(N)39 b FF(such)26 b(that)g(all)i(pr)n(oducts)d (of)i(matrices)f Fz(A)2008 3217 y Fs(1)2048 3203 y Fz(A)2116 3217 y Fs(2)2171 3203 y Fz(:)15 b(:)g(:)h(A)2360 3217 y Fr(r)2425 3203 y FF(in)26 b Fz(M)37 b FF(of)27 b(length)e Fz(r)k FF(have)e(a)g(coef-)300 3373 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b(lend)g(themselves)f(well)h (to)g(studying)f(mor)n(e)h(general)g(set-chains,)h(wher)n(e)f Fz(M)37 b FF(is)28 b(not)f(an)300 4219 y(interval.)300 4528 y Fo(1.3.2)99 b(Pr)n(oduct)25 b(of)g(Sets)g(of)f(Matrices)300 4761 y FF(While)k(Hart\002el)f(used)e(mostly)i(matrix-theor)n(etic)g (techniques)f(in)i(the)e(analysis)i(of)f(Markov)300 4931 y(set-chains,)20 b(Saks)f(and)g(Condon)g([34)q(])h(studied)e(the)g(pr)n (oducts)h(of)g(sets)f(of)i(stochastic)f(matrices)300 5100 y(using)k(graph-theor)n(etic)g(techniques.)29 b(They)23 b(showed)e(that)j(for)f(any)h(\002nite)f(set)g(of)g(stochastic)1927 5349 y(4)p eop %%Page: 5 10 5 9 bop 300 391 a Fz(A)p FF(,)36 b(ther)n(e)d(exist)g(a)h Fz(k)48 b FD(2)e Fm(N)6 b FF(,)42 b(such)34 b(that)f(the)g(sequence)f (of)h(sets)f FA(\()p Fz(A)2663 358 y Fr(k)r(i)2730 391 y FA(\))2765 405 y Fr(i)p Fn(2)p Fl(N)2922 391 y FF(conver)n(ges)h(in)h (the)300 561 y(Hausdorf)n(f)29 b(metric.)50 b(This)29 b(r)n(esult)g(is)h(quite)f(str)n(ong)f(in)j(the)e(sense)e(that)j(it)g (does)e(not)h(r)n(ely)g(on)300 730 y(any)23 b(spectral)f(pr)n(operties) e(or)j(the)f(matrices,)g(nor)g(on)h(any)f(pr)n(operties)f(of)h(the)g (set)g Fz(A)p FF(.)300 1043 y Fo(1.3.3)99 b(Random)25 b(W)n(alk)h(with)e(Internal)i(States)300 1276 y FF(The)g(theory)f(of)i (random)g(walk)g(with)g(internal)h(states)d(was)i(\002rst)f(intr)n (oduced)g(by)h(Sinai)h([37)q(])300 1446 y(and)e(later)f(studied)f(in)i (depth)e(by)h(Kr)8 b(\264)-38 b(amli)27 b(and)f(Sz)8 b(\264)-38 b(asz)26 b([28)q(,)g(29,)g(27)q(].)37 b(The)24 b(model)h(was)h(intr)n(o-)300 1615 y(duced)19 b(as)h(a)g(tool)g(in)g (the)f(study)g(of)h(the)f(behaviour)h(of)g(a)h(class)f(of)g(physical)g (pr)n(ocesses)e(known)300 1784 y(as)31 b(the)f(Lor)n(entz)g(pr)n (ocess.)50 b(In)31 b(their)f(formulation,)j(a)e(random)g(walk)g(with)g (internal)g(states)300 1954 y Fz(X)375 1968 y Fr(t)430 1954 y FA(=)25 b(\()p Fz(Z)623 1968 y Fr(t)653 1954 y Fz(;)15 b(Y)746 1968 y Fr(t)776 1954 y FA(\))21 b FF(has)g(both)g(a)g (random)g(walk)g(component)e 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b(he)d(showed)g(that)h(the)g(\223backwar)n(d\224)i(pr)n(oduct)d Fz(B)2662 1083 y Fr(r)2736 1069 y FA(=)35 b Fz(A)2910 1083 y Fr(r)2948 1069 y Fz(B)3017 1083 y Fr(r)r Fn(\000)p Fs(1)3174 1069 y FF(is)28 b(str)n(ongly)300 1238 y(er)n(godic,)d(which) h(can)g(be)g(interpr)n(eted)d(as)i(the)g(existence)f(of)h(a)h(unique)f (limit)i Fz(B)3103 1253 y Fs(lim)3230 1238 y FF(such)e(that)300 1408 y FD(j)p Fz(B)394 1423 y Fs(lim)517 1408 y FD(\000)19 b Fz(B)676 1422 y Fr(r)714 1408 y FD(j)26 b Fz(<)f(")e FF(for)g(any)f Fz(")k(>)f FA(0)e FF(and)g Fz(r)i FF(suf)n(\002ciently)d (lar)n(ge.)300 1781 y Ft(1.4)119 b(Or)n(ganization)300 2048 y FF(This)22 b(thesis)g(is)g(or)n(ganized)h(in)g(the)f(following)h (way:)436 2292 y FD(\017)46 b FF(Chapter)32 b(2)h(covers)f(the)g (backgr)n(ound)g(material,)k(and)c(rigor)n(ously)g(de\002nes)e(random) 527 2462 y(set-walk)22 b(and)h(r)n(elated)f(concepts.)436 2706 y FD(\017)46 b FF(Chapter)17 b(3)h(looks)e(at)i(the)e(case)i(when) 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Fs(\()p Fr(t)p Fs(\))1918 4630 y FA(=)k Fz(P)2085 4592 y Fr(t)2137 4630 y Fz(:)1264 b FF(\(2.5\))583 4858 y(Consequently)-10 b(,)21 b(the)h(pr)n(obability)h FB(P)p FA(\()p Fz(X)1960 4872 y Fr(t)2016 4858 y FA(=)i Fz(i)p FA(\))e FF(satis\002es)1619 5100 y FB(P)p FA(\()p Fz(X)1787 5114 y Fr(t)1842 5100 y FA(=)i Fz(i)p FA(\))h(=)f Fz(\026P)2252 5062 y Fr(t)3426 5100 y FF(\(2.6\))1927 5349 y(8)p eop %%Page: 9 14 9 13 bop 300 391 a FF(wher)n(e)21 b Fz(\026)k FA(=)g(\()p FB(P)p FA(\()p Fz(X)951 405 y Fs(0)1016 391 y FA(=)g Fz(i)p FA(\)\))1213 405 y Fr(i)p Fn(2)p Fr(S)1358 391 y FF(is)d(the)e Fw(initial)j(distribution)h(vector)p FF(,)f(which)f(is)g(the)f(r)n(ow)g(vector)g(of)300 561 y(initial)k(pr)n(obabilities)e(for)g(the)f(chain.)300 805 y Fp(De\002nition)28 b(2.4)i(\(Stochastic)g(and)f(substochastic)h (matrices\).)48 b FF(Let)28 b Fz(M)47 b FA(=)37 b([)p Fz(m)3102 819 y Fr(ij)3162 805 y FA(])30 b FF(be)f(a)g(non-)300 974 y(negative)24 b(matrix.)35 b(If)25 b Fz(M)34 b FF(satis\002es)1539 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Fz(d)g FD(2)g Fm(Z)p Fw(.)1927 5349 y FF(9)p eop %%Page: 10 15 10 14 bop 300 391 a Fo(2.2.1)99 b(Gambler)t(')-5 b(s)22 b(Ruin)k(and)g(Stopping)f(T)-5 b(ime)300 625 y FF(While)29 b(the)g(simple)g(random)g(walk)g(is)h(a)f(pr)n(ocess)f(that)h(takes)f (values)h(on)g(the)f(all)j(of)e(the)f(in-)300 794 y(tegers,)33 b(we)f(can)i(r)n(estrict)e(a)h(random)g(walk)g(to)f(a)h(particular)h (interval)g(of)e(integers,)i(which)300 963 y(traditionally)23 b(is)g(the)f(interval)h FA(0)p Fz(;)15 b(:)g(:)g(:)i(;)e(N)10 b FF(.)300 1204 y Fp(Notation.)41 b FF(W)-8 b(e)22 b(shall)h(denote)e (the)h(integer)f(interval)i Fz(a;)15 b(:)g(:)g(:)i(;)e(b)23 b FF(as)g FA([)p Fz(a)i FA(:)h Fz(b)p FA(])p FF(.)i(Mor)n(e)22 b(formally)-10 b(,)1384 1462 y FA([)p Fz(a)26 b FA(:)f Fz(b)p FA(])1645 1410 y Fz(:)1623 1462 y FA(=)g FD(f)p Fz(i)h FD(2)f Fm(Z)18 b FD(j)23 b Fz(a)i FD(\024)g Fz(i;)15 b(i)26 b FD(\024)f Fz(b)p FD(g)911 b FF(\(2.9\))300 1719 y(wher)n(e)22 b Fz(a;)15 b(b)26 b FD(2)e Fm(Z)p FF(.)g(Thus)e FA([)p Fz(a)j 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/Times-Italic ff 360.00 scf sf 2550 3975 m gs 1 -1 sc (al) col0 sh gr /Times-Italic ff 360.00 scf sf 6900 3975 m gs 1 -1 sc (cl) col0 sh gr /Times-Italic ff 360.00 scf sf 600 6450 m gs 1 -1 sc (bl) col0 sh gr /Times-Italic ff 360.00 scf sf 4500 6450 m gs 1 -1 sc (dl) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 438 2750 a currentpoint currentpoint translate 1 .40 div 1 .40 div scale neg exch neg exch translate 438 2750 a 933 2946 a FF(Figur)n(e)f(2.1:)29 b(Concatenating)22 b(two)g(characteristic)i(tuples)583 3215 y(Let)g FA(\()p Fz(\030)812 3229 y Fr(t)841 3215 y FA(\))29 b(=)g Fz(W)13 b FA(\()p Fz(s)1182 3182 y Fn(0)1204 3215 y Fz(s)p FA(\))25 b FF(be)f(the)g(walk)h(instance)f(of)g Fz(s)2300 3182 y Fn(0)2323 3215 y Fz(s)p FF(,)h(and)f(the)g(stopping)e (times)i Fz(T)3420 3229 y Fs(0)3460 3215 y FF(,)g Fz(T)3560 3229 y Fs(1)300 3385 y FF(and)f Fz(T)530 3399 y Fs(2)592 3385 y FF(be)f(de\002ned)f(as)i(follows:)1119 3645 y Fz(T)1172 3659 y Fs(0)1236 3645 y FA(=)i(inf)6 b FD(f)p Fz(t)26 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Fz(e)p FD(g)30 b FF(wher)n(e)e Fz(e)g FF(is)h(the)300 2788 y(sole)21 b(element)f(of)i(the)f(state)f(space.)27 b(When)21 b(ther)n(e)g(is)g(only)g(a)h(single)g(internal)f(state,)g (all)i(transi-)300 2958 y(tion)h(pr)n(obability)h(matrices)f(can)h(be)e (r)n(educed)g(to)g(scalar)i(transition)f(pr)n(obabilities.)33 b(Hence)23 b(a)300 3127 y(transition)f(triple)h FB(T)f FF(is)h(in)g FA([0)p Fz(;)15 b FA(1])1404 3094 y Fs(3)1445 3127 y FF(,)23 b(and)f(a)i(characteristic)f(tuple)f Fz(')h FF(is)g(in)g FA([0)p Fz(;)15 b FA(1])2973 3094 y Fs(4)3014 3127 y FF(.)300 3500 y Ft(3.1)119 b(Main)29 b(Result)300 3768 y FF(The)17 b(main)i(r)n(esult)e(of)h(this)f(thesis)g(concerns)g (the)f(long-r)o(un)i(behaviour)g(of)g(random)f(set-walks.)300 3937 y(Since,)28 b(for)g(the)e(majority)i(of)g(transition)f(sets)f FD(T)c FF(,)29 b(the)d(external)h(states)f FA([1)35 b(:)f Fz(n)p FA(])g FD(\032)g FA([0)h(:)f Fz(n)24 b FA(+)f(1])300 4106 y FF(of)33 b(the)f(walk)i(instance)f Fz(W)13 b FA(\()p Fz(s)p FA(\))33 b FF(of)g Fz(s)44 b FD(2)g Fm(S)1774 4125 y Fs([)o(1:)p Fr(n)p Fs(])1909 4106 y FA(\()p FD(T)23 b FA(\))33 b FF(ar)n(e)h(transient,)g(the)f(stationary)f(distribu-)300 4276 y(tion)27 b(is)g(governed)e(solely)h(by)h(the)g(pr)n(obabilities)h (given)f(in)g(the)f(characteristic)j(tuple)d(of)h(the)300 4445 y(walk)38 b(instance.)74 b(Hence,)41 b(looking)c(at)i(the)e(set)g (of)h(characteristic)h(pr)n(obabilities)g(over)f(all)300 4615 y Fz(s)i FD(2)h Fm(S)541 4633 y Fs([1:)p Fr(n)p Fs(])676 4615 y FA(\()p FD(T)23 b FA(\))32 b FF(pr)n(ovides)e(some)g (insight)h(into)g(the)f(stationary)g(pr)n(operties)f(of)i(the)f(random) 300 4784 y(set-walk)22 b FD(W)7 b FA(\()p Fm(S)852 4802 y Fs([)o(1:)p Fr(n)p Fs(])987 4784 y FA(\()p FD(T)23 b FA(\)\))p FF(.)28 b(W)-8 b(e)22 b(discover)n(ed)f(that,)h(in)h(fact,) g(these)d(sets)h(of)h(characteristic)i(pr)n(ob-)300 4953 y(abilities)i(conver)n(ge)e(as)h(we)f(lengthen)f(the)h(random)g (set-walk)g(by)h(letting)f Fz(n)k FD(!)h(1)p FF(.)34 b(In)24 b(other)1905 5349 y(23)p eop %%Page: 24 29 24 28 bop 300 391 a FF(wor)n(ds,)29 b(the)f(long-r)o(un)g(behaviour)h (of)g(a)g(random)f(set-walk)h(r)n(elative)g(to)f(its)g(endpoints)f(be-) 300 561 y(comes)g(incr)n(easingly)h(independent)d(of)i(the)g(length)f (of)i(the)e(set-walk)h(as)h(it)f(appr)n(oaches)g(in-)300 730 y(\002nity)-10 b(.)300 966 y Fp(De\002nition)16 b(3.1)i (\(Characteristic)j(tuple)d(set\).)34 b FF(De\002ne)17 b(the)f(4-tuple)h FA(\010)2726 980 y Fr(n)2773 966 y FA(\()p FD(T)23 b FA(\))i(=)g FD(h)p FA(A)3140 980 y Fr(n)3188 966 y Fz(;)15 b FA(B)3292 980 y Fr(n)3340 966 y Fz(;)g FA(\000)3437 980 y Fr(n)3484 966 y Fz(;)g FA(\001)3600 980 y Fr(n)3647 966 y FD(i)300 1135 y FF(as)23 b(the)f Fw(characteristic)k(tuple)e(set)f FF(of)g Fm(S)1560 1154 y Fs([1:)p Fr(n)p Fs(])1696 1135 y FA(\()p FD(T)g FA(\))p FF(,)f(wher)n(e)1139 1386 y FA(A)1207 1400 y Fr(n)1279 1386 y FA(=)j Fz(a)p FA([)p Fm(S)1505 1405 y Fs([1:)p Fr(n)o Fs(])1640 1386 y FA(\()p FD(T)e FA(\)])49 b(=)25 b FD(f)p Fz(a)p FA([)p Fz(s)p FA(])e FD(j)g Fz(s)i FD(2)g Fm(S)2420 1405 y Fs([1:)p Fr(n)p Fs(])2555 1386 y FA(\()p FD(T)e FA(\))p FD(g)1143 1581 y FA(B)1207 1595 y Fr(n)1279 1581 y FA(=)i Fz(b)p FA([)p Fm(S)1496 1599 y Fs([1:)p Fr(n)o Fs(])1631 1581 y FA(\()p FD(T)e FA(\)])58 b(=)25 b FD(f)p Fz(b)p FA([)p Fz(s)p FA(])e FD(j)g Fz(s)i FD(2)g Fm(S)2411 1599 y Fs([1:)p Fr(n)p Fs(])2546 1581 y FA(\()p FD(T)e FA(\))p FD(g)1150 1775 y FA(\000)1207 1789 y Fr(n)1279 1775 y FA(=)i Fz(c)p FA([)p Fm(S)1496 1794 y Fs([1:)p Fr(n)p Fs(])1631 1775 y FA(\()p FD(T)e FA(\)])58 b(=)25 b FD(f)p Fz(c)p FA([)p Fz(s)p FA(])e FD(j)g Fz(s)i FD(2)g Fm(S)2412 1794 y Fs([)o(1:)p Fr(n)p Fs(])2547 1775 y FA(\()p FD(T)e FA(\))p FD(g)1131 1969 y FA(\001)1207 1983 y Fr(n)1279 1969 y FA(=)i Fz(d)p FA([)p Fm(S)1504 1988 y Fs([1:)p Fr(n)p Fs(])1639 1969 y FA(\()p FD(T)e FA(\)])50 b(=)25 b FD(f)p Fz(d)p FA([)p Fz(s)p FA(])e FD(j)g Fz(s)i FD(2)g Fm(S)2420 1988 y Fs([)o(1:)p Fr(n)p Fs(])2555 1969 y FA(\()p FD(T)e FA(\))p FD(g)p Fz(:)300 2220 y FF(Wher)n(eas)37 b(a)i(characteristic)g(tuple)f(describes)f(a)i (random)f(walk)g(in)h(a)g(nonhomogeneous)300 2390 y(bounded)32 b(envir)n(onment)h(that)h(is)f(governed)f(by)i(a)g(particular)h (strategy)c Fz(s)p FF(,)36 b(a)e(characteris-)300 2559 y(tic)h(tuple)f(set)g(describes)g(the)g(random)h(set-walk)f FD(W)7 b FA(\()p Fm(S)2310 2578 y Fs([1:)o Fr(n)p Fs(])2445 2559 y FA(\()p FD(T)23 b FA(\)\))36 b FF(governed)d(by)h(the)g(set)g (of)300 2728 y(strategies)21 b Fm(S)761 2747 y Fs([1:)p Fr(n)o Fs(])896 2728 y FA(\()p FD(T)i FA(\))p FF(.)300 2964 y Fp(Theorem)28 b(3.2)i(\(Main)f(Theorem\).)44 b Fw(If)30 b(the)f(internal)h(state)g(space)g Fz(E)k Fw(contains)29 b(only)g(a)g(single)h(el-)300 3134 y(ement,)f(then,)g(for)g(any)f (non-empty)g(transition)i(set)e FD(T)23 b Fw(,)28 b(the)h(sequence)f (of)h(characteristic)i(tuple)e(sets)300 3303 y FA(\(\010)401 3317 y Fr(n)448 3303 y FA(\()p FD(T)23 b FA(\)\))g Fw(conver)n(ges)j (componentwise)e(in)e(the)i(Hausdorff)h(metric.)583 3539 y FF(This)30 b(theor)n(em)f(statement)f(is)i(in)g(fact)h(slightly)e (str)n(onger)f(than)i(the)f(claim)j(made)e(in)300 3709 y(the)f(intr)n(oductory)f(paragraph)i(of)f(this)g(section:)42 b(we)29 b(will)i(pr)n(ove)e(the)g(conver)n(gence)f(of)i(the)300 3878 y(characteristic)25 b(tuple)e(sets)f(even)h(when)g(the)g (individual)i(walk)f(instances)f Fz(W)13 b FA(\()p Fz(s)p FA(\))24 b FF(contain)g(r)n(e-)300 4047 y(curr)n(ent)e(states)f(other)h (than)g(the)g(endpoints,)e(which)k(is)e(the)g(case)h(when)f FA(\(0)p Fz(;)15 b FA(1)p Fz(;)g FA(0\))28 b FD(2)d(T)d FF(.)300 4359 y Fo(3.1.1)99 b(Overview)23 b(of)i(the)f(Pr)n(oof)300 4592 y FF(In)k(the)f(following)h(sections,)g(we)f(pr)n(esent)f(the)h (pr)n(oofs)g(of)h(the)f(Main)i(Theor)n(em)e(for)h(various)300 4761 y(dif)n(fer)n(ent)e(cases,)h(fr)n(om)f(which)h(we)f(will)i(then)d (be)h(able)h(to)f(piece)g(together)d(the)j(pr)n(oof)g(of)g(the)300 4931 y(theor)n(em)e(in)i(its)f(most)f(general)g(context.)35 b(The)24 b(special)i(cases)e(that)h(we)g(will)h(consider)e(ar)n(e)i(as) 300 5100 y(follows:)1905 5349 y(24)p eop %%Page: 25 30 25 29 bop 414 391 a FF(1.)45 b(The)19 b(case)h(wher)n(e)e(all)j(the)e (transition)g(triples)g(in)h(the)f(transition)g(set)f(ar)n(e)i (\223left-leaning\224,)527 561 y(which)30 b(is)f(to)g(say)g Fz(\015)5 b FA([)p FB(T)p FA(])38 b FD(\024)f FA(1)p Fz(=)p FA(2)31 b FF(for)e(each)h FB(T)37 b FD(2)g(T)23 b FF(.)48 b(The)28 b(conver)n(gence)g(pr)n(oof)h(for)g(this)527 730 y(case)23 b(is)g(pr)n(esented)d(in)j(Section)f(3.3.)414 960 y(2.)45 b(The)25 b(case)h(wher)n(e)e(all)k(the)c(triples)h(in)h (the)f(transition)g(set)g(ar)n(e)h(\223right-leaning\224,)h(which)527 1129 y(is)i(to)f(say)g Fz(\015)5 b FA([)p FB(T)p FA(])37 b Fz(>)f FA(1)p Fz(=)p FA(2)30 b FF(for)f(each)g FB(T)36 b FD(2)g(T)23 b FF(.)46 b(The)28 b(conver)n(gence)f(pr)n(oof)i(for)f (this)g(case)h(is)527 1299 y(pr)n(esented)20 b(in)j(Section)f(3.4.)414 1529 y(3.)45 b(The)30 b(case)g(wher)n(e)f(ther)n(e)h(is)g(a)h(mix)g(of) f(left-)h(and)f(right-leaning)g(transition)g(triples)g(in)527 1698 y(the)23 b(transition)g(set.)29 b(The)23 b(conver)n(gence)f(pr)n (oof)h(for)g(this)g(case)h(is)f(pr)n(esented)e(in)j(Section)527 1867 y(3.5.)414 2097 y(4.)45 b(The)35 b(bor)n(der)f(cases)h(involving)i (the)d(triples)h FA(\(1)p Fz(;)15 b FA(0)p Fz(;)g FA(0\))p FF(,)41 b FA(\(0)p Fz(;)15 b FA(0)p Fz(;)g FA(1\))p FF(,)41 b(and)35 b FA(\(0)p Fz(;)15 b FA(1)p Fz(;)g FA(0\))39 b FF(\227)c(the)527 2267 y(conver)n(gence)d(pr)n(oofs)g(for)g(which)h (ar)n(e)g(pr)n(esented)d(in)j(Sections)f(3.6,)k(3.7)d(and)g(3.8)g(r)n (e-)527 2436 y(spectively)-10 b(.)300 2645 y(Note)22 b(that,)h(in)h(these)e(sections,)g(the)h(main)h(focus)f(will)i(be)f (establishing)f(the)g(conver)n(gence)f(of)300 2814 y(the)e(sets)f FA(\000)675 2828 y Fr(n)721 2814 y FF(.)28 b(The)20 b(conver)n(gence)f (of)i FA(\010)1635 2828 y Fr(n)1702 2814 y FF(will)h(in)f(fact)g (follow)g(quite)f(naturally)-10 b(,)22 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y(cerned)c(with)h (random)f(set-walks)g(wher)n(e)g FD(j)p Fz(E)5 b FD(j)29 b FA(=)e(1)p FF(,)d(the)f(function)h FB(T)p FA(\()p Fz(x)p FA(\))h FF(allows)f(us)f(to)h(spec-)300 5100 y(ify)29 b(transition)e(triples)h(by)g(their)g(transition)g(pr)n(obabilities)h (in)g(a)g(straightforwar)n(d)e(manner)-7 b(.)1905 5349 y(25)p eop %%Page: 26 31 26 30 bop 300 391 a FF(Mor)n(eover)-7 b(,)20 b(this)g(de\002nition)g (makes)g(it)h(simpler)g(to)f(divide)h(the)f(pr)n(oof)g(of)h(the)e(Main) j(Theor)n(em)300 561 y(into)g(special)i(cases)e(based)g(on)g(the)g (\223leanings\224)h(of)g(the)f(triples.)300 868 y Fo(3.2.2)99 b(Basic)24 b(Conver)n(gence)h(Results)300 1101 y FF(The)30 b(following)h(ar)n(e)g(thr)n(ee)f(basic)i(r)n(esults)d(r)n(egar)n(ding) i(the)e(conver)n(gence)h(of)h(a)g(sequence)e(of)300 1271 y(characteristic)21 b(pr)n(obabilities)g(associated)e(with)h(a)g (strategy)e(sequence.)25 b(These)18 b(r)n(esults)h(ar)n(e)h(to)300 1440 y(be)e(the)e(building)j(blocks)e(with)h(which)g(we)f(constr)o(uct) g(the)g(subpr)n(oofs)f(of)i(the)f(Main)h(Theor)n(em.)300 1658 y Fp(Proposition)j(3.4.)42 b Fw(If)23 b FB(T)p FA(\()p Fz(x)p FA(\))i FD(2)g(T)45 b Fw(wher)n(e)24 b Fz(x)h FD(\024)g FA(1)p Fz(=)p FA(2)p Fw(,)f(and)f Fz(U)10 b FA(\()p FB(T)p FA(\()p Fz(x)p FA(\)\))26 b(=)f(\()p Fz(s)2731 1672 y Fr(n)2778 1658 y FA(\))e Fw(is)g(a)g(uniform)f(strategy)300 1827 y(sequence,)h(then)g(ther)n(e)i(is)f(a)f(limit)g Fz(c)1439 1841 y Fn(1)1537 1827 y Fw(such)g(that)h Fz(c)p FA([)p Fz(s)2004 1841 y Fr(n)2051 1827 y FA(])i FD(")g Fz(c)2212 1841 y Fn(1)2287 1827 y Fw(,)c(and)h Fz(d)p FA([)p Fz(s)2606 1841 y Fr(n)2653 1827 y FA(])j FD(#)g FA(0)p Fw(.)300 2066 y(Pr)n(oof.)46 b FF(By)22 b(the)g(gambler)7 b('s)23 b(r)o(uin)g(r)n(esult)f(\(2.12\))r(,)1433 2400 y Fz(c)p FA([)p Fz(s)1540 2414 y Fr(n)1587 2400 y FA(])j(=)1733 2159 y Fq(8)1733 2241 y(>)1733 2268 y(>)1733 2295 y(<)1733 2459 y(>)1733 2486 y(>)1733 2514 y(:)1863 2253 y FA(\()1908 2219 y Fk(1)p Fi(\000)p Fj(x)p 1908 2232 114 3 v 1947 2273 a(x)2031 2253 y FA(\))2066 2206 y Fj(n)2109 2247 y Fn(\000)p Fs(1)p 1824 2283 414 4 v 1824 2366 a FA(\()1869 2332 y Fk(1)p Fi(\000)p Fj(x)p 1869 2345 114 3 v 1909 2386 a(x)1993 2366 y FA(\))2028 2319 y Fj(n)p Fk(+1)2147 2360 y Fn(\000)p Fs(1)2361 2304 y FF(if)f Fz(x)h(<)g FA(1)p Fz(=)p FA(2)1814 2532 y(1)c FD(\000)f Fz(x=n)238 b FF(if)24 b Fz(x)h FA(=)g(1)p Fz(=)p FA(2)1148 2747 y Fe(\))53 b FA(lim)1234 2802 y Fr(n)p Fn(!1)1433 2747 y Fz(c)p FA([)p Fz(s)1540 2761 y Fr(n)1587 2747 y FA(])25 b(=)1822 2686 y Fz(x)p 1743 2726 209 4 v 1743 2809 a FA(1)c FD(\000)f Fz(x)300 3001 y FF(and)1425 3336 y Fz(d)p FA([)p Fz(s)1540 3350 y Fr(n)1587 3336 y FA(])25 b(=)1733 3095 y Fq(8)1733 3177 y(>)1733 3204 y(>)1733 3231 y(<)1733 3395 y(>)1733 3422 y(>)1733 3449 y(:)1929 3158 y Fk(1)p Fi(\000)p Fj(x)p 1929 3171 114 3 v 1968 3212 a(x)2052 3186 y Fn(\000)p Fs(1)p 1824 3219 414 4 v 1824 3302 a FA(\()1869 3268 y Fk(1)p Fi(\000)p Fj(x)p 1869 3281 114 3 v 1909 3322 a(x)1993 3302 y FA(\))2028 3255 y Fj(n)p Fk(+1)2147 3296 y Fn(\000)p Fs(1)2361 3240 y FF(if)f Fz(x)h(<)g FA(1)p Fz(=)p FA(2)1814 3467 y Fz(x=n)395 b FF(if)24 b Fz(x)h FA(=)g(1)p Fz(=)p FA(2)1140 3683 y Fe(\))54 b FA(lim)1226 3737 y Fr(n)p Fn(!1)1425 3683 y Fz(d)p FA([)p Fz(s)1540 3697 y Fr(n)1587 3683 y FA(])25 b(=)g(0)p FF(,)e(since)g Fz(x)i FD(\024)g FA(1)p Fz(=)p FA(2)p 3534 3912 4 62 v 3538 3854 55 4 v 3538 3912 V 3592 3912 4 62 v 300 4150 a Fp(Proposition)18 b(3.5.)36 b Fw(Let)19 b FA(\()p Fz(s)1189 4164 y Fr(i)1217 4150 y FA(\))g Fw(be)h(a)f(monotone)h(strategy)i(sequence)d(on)g FD(T)42 b Fw(wher)n(e)20 b(the)g(sequence)g FA(\()p Fz(c)p FA([)p Fz(s)3511 4164 y Fr(i)3539 4150 y FA(]\))300 4320 y Fw(conver)n(ges.)31 b(Then)22 b(for)i(any)f Fz(s)1245 4287 y Fn(0)1293 4320 y FD(2)i Fm(S)o FA(\()p FD(T)16 b FA(\))p Fw(,)23 b(the)h(sequence)f FA(\()p Fz(c)p FA([)p Fz(s)2238 4334 y Fr(i)2267 4320 y Fz(s)2310 4287 y Fn(0)2333 4320 y FA(]\))g Fw(also)i(conver)n(ges.)300 4558 y(Pr)n(oof.)46 b FF(Let)22 b Fz(')774 4574 y Fr(s)807 4555 y Fi(0)859 4558 y FA(=)j FD(h)p Fz(a;)15 b(b;)g(c;)g(d)p FD(i)p FF(.)30 b(Then)1025 4787 y Fz(c)p FA([)p Fz(s)1132 4801 y Fr(i)1161 4787 y Fz(s)1204 4750 y Fn(0)1227 4787 y FA(])25 b(=)g Fz(c)c FA(+)1561 4726 y Fz(a)i(c)p FA([)p Fz(s)1739 4740 y Fr(i)1767 4726 y FA(])g Fz(d)p 1533 4766 357 4 v 1533 4849 a FA(1)e FD(\000)f Fz(c)p FA([)p Fz(s)1797 4863 y Fr(i)1825 4849 y FA(])p Fz(b)1277 5030 y FA(=)25 b Fz(f)10 b FA(\()p Fz(c)p FA([)p Fz(s)1570 5044 y Fr(i)1598 5030 y FA(]\))205 b FF(wher)n(e)22 b Fz(f)10 b FA(\()p Fz(x)p FA(\))25 b(=)g Fz(c)c FA(+)2645 4969 y Fz(axd)p 2595 5009 248 4 v 2595 5092 a FA(1)f FD(\000)g Fz(xb)2852 5030 y FF(.)1905 5349 y(26)p eop %%Page: 27 32 27 31 bop 300 391 a FF(By)24 b(the)f(continuity)h(of)h Fz(f)33 b FF(on)25 b FA([0)p Fz(;)15 b FA(1])p FF(,)26 b(if)f Fz(c)p FA([)p Fz(s)1747 405 y Fr(i)1776 391 y FA(])g FF(conver)n(ges)e(as)h Fz(i)29 b FD(!)g(1)p FF(,)c(then)e Fz(f)10 b FA(\()p Fz(c)p FA([)p Fz(s)3082 405 y Fr(i)3110 391 y FA(]\))25 b FF(conver)n(ges)300 561 y(as)e(well.)p 3534 561 4 62 v 3538 503 55 4 v 3538 561 V 3592 561 4 62 v 300 805 a Fp(Corollary)31 b(3.6.)47 b Fw(Suppose)32 b Fz(s)40 b FD(2)h Fm(S)n FA(\()p FD(T)17 b FA(\))p Fw(,)33 b(and)f FA(\()p Fz(l)1906 819 y Fr(i)1934 805 y FA(\))42 b(=)e Fz(U)10 b FA(\()p FB(T)p FA(\()p Fz(l)r FA(\)\))32 b Fw(is)g(a)f(uniform)g(strategy)j(sequence)300 974 y(wher)n(e)24 b Fz(l)j(<)e FA(1)p Fz(=)p FA(2)f Fw(and)g FB(T)p FA(\()p Fz(l)r FA(\))h FD(2)g(T)e Fw(.)28 b(Then)22 b(the)i(sequence)g FA(\()p Fz(c)p FA([)p Fz(l)2217 988 y Fr(i)2246 974 y Fz(s)p FA(]\))f Fw(conver)n(ges.)300 1218 y(Pr)n(oof.)46 b FF(By)25 b(Pr)n(oposition)g(3.4,)i Fz(c)p FA([)p Fz(l)1442 1232 y Fr(i)1471 1218 y FA(])f FF(conver)n(ges)e(as)i Fz(i)31 b FD(!)g(1)p FF(,)26 b(and)f(hence)g(it)h(follows)g(that)f Fz(c)p FA([)p Fz(l)3503 1232 y Fr(i)3532 1218 y Fz(s)p FA(])300 1388 y FF(conver)n(ges)c(as)i(well,)g(as)g Fz(i)i FD(!)g(1)p FF(,)e(by)f(Pr)n(oposition)g(3.5.)p 3534 1388 V 3538 1330 55 4 v 3538 1388 V 3592 1388 4 62 v 300 1701 a Fo(3.2.3)99 b(The)24 b(Comparison)h(Lemma)300 1934 y FF(Just)17 b(as)g(important)g(as)g(the)f(pr)n(evious)h(r)n(esults)f (on)h(conver)n(gence)f(is)h(the)g(Comparison)g(Lemma,)300 2103 y(so)24 b(named)g(because)g(it)h(allows)f(us)g(to)g(state)f (inequalities)i(about)f(two)g(characteristic)h(pr)n(oba-)300 2273 y(bilities)k Fz(c)p FA([)p Fz(s)p FA(])g FF(and)f Fz(c)p FA([)p Fz(s)1046 2240 y Fn(0)1069 2273 y FA(])g FF(based)g(solely)f(on)h(the)f(composition)g(of)h(the)g(strategies)e Fz(s)h FF(and)h Fz(s)3426 2240 y Fn(0)3449 2273 y FF(.)44 b(In)300 2442 y(essence,)32 b(ther)n(e)e(is)i(a)g(partial)h(or)n (dering)d(of)i(the)f(strategies)e(in)j Fm(S)o FA(\()p FD(T)17 b FA(\))32 b FF(that)f(is)h(similar)h(in)f(na-)300 2611 y(tur)n(e)g(to)g(\227)h(but)f(not)g(the)g(same)h(as)f(\227)h(a)g (typographical)g(or)n(dering:)47 b(for)32 b Fz(s;)15 b(s)3071 2578 y Fn(0)3138 2611 y FD(2)43 b Fm(S)3299 2630 y Fs([)o(1:)p Fr(n)p Fs(])3434 2611 y FA(\()p FD(T)23 b FA(\))p FF(,)300 2781 y Fz(c)p FA([)p Fz(s)407 2748 y Fn(0)430 2781 y FA(])j FD(\024)f Fz(c)p FA([)p Fz(s)p FA(])e FF(if)h Fz(\015)5 b FA([)p Fz(s)932 2748 y Fn(0)955 2781 y FA(\()p Fz(z)t FA(\)])26 b FD(\024)f Fz(\015)5 b FA([)p Fz(s)p FA(\()p Fz(z)t FA(\)])24 b FF(for)f(each)g Fz(z)29 b FD(2)c FA([1)h(:)g Fz(n)p FA(])p FF(.)300 3025 y Fp(Proposition)c(3.7.)41 b Fw(Let)23 b Fz(s)1167 3040 y Fr(l)1193 3025 y Fz(;)15 b(s)1276 3039 y Fr(r)1314 3025 y Fz(;)g(s)25 b FD(2)g Fm(S)o FA(\()p FD(T)17 b FA(\))p Fw(.)28 b(If)23 b Fz(c)p FA([)p Fz(s)1938 3040 y Fr(l)1964 3025 y FA(])j Fz(<)f(c)p FA([)p Fz(s)2218 3039 y Fr(r)2256 3025 y FA(])p Fw(,)e(then)g Fz(c)p FA([)p Fz(s)2618 3040 y Fr(l)2644 3025 y Fz(s)p FA(])i Fz(<)g(c)p FA([)p Fz(s)2940 3039 y Fr(r)2978 3025 y Fz(s)p FA(])p Fw(.)300 3269 y(Pr)n(oof.)46 b FF(Let)22 b Fz(')774 3283 y Fr(s)836 3269 y FA(=)j FD(h)p Fz(a;)15 b(b;)g(c;)g(d)p FD(i)p FF(.)31 b(Then)1527 3530 y Fz(c)p FA([)p Fz(s)1634 3545 y Fr(l)1660 3530 y Fz(s)p FA(])25 b(=)g Fz(c)20 b FA(+)2048 3468 y Fz(a)j(c)p FA([)p Fz(s)2226 3483 y Fr(l)2252 3468 y FA(])p Fz(d)p 2009 3509 355 4 v 2009 3592 a FA(1)e FD(\000)f Fz(c)p FA([)p Fz(s)2273 3607 y Fr(l)2299 3592 y FA(])p Fz(b)1515 3777 y(c)p FA([)p Fz(s)1622 3791 y Fr(r)1660 3777 y Fz(s)p FA(])25 b(=)g Fz(c)20 b FA(+)2048 3716 y Fz(a)j(c)p FA([)p Fz(s)2226 3730 y Fr(r)2264 3716 y FA(])p Fz(d)p 2009 3757 366 4 v 2009 3840 a FA(1)e FD(\000)f Fz(c)p FA([)p Fz(s)2273 3854 y Fr(r)2311 3840 y FA(])p Fz(b)300 4038 y FF(and)1119 4146 y Fz(a)i(c)p FA([)p Fz(s)1296 4161 y Fr(l)1323 4146 y FA(])p Fz(d)p 1080 4187 355 4 v 1080 4270 a FA(1)e FD(\000)g Fz(c)p FA([)p Fz(s)1343 4285 y Fr(l)1370 4270 y FA(])p Fz(b)1469 4208 y(<)1614 4146 y(a)j(c)p FA([)p Fz(s)1792 4160 y Fr(r)1830 4146 y FA(])p Fz(d)p 1575 4187 366 4 v 1575 4270 a FA(1)e FD(\000)f Fz(c)p FA([)p Fz(s)1839 4284 y Fr(r)1877 4270 y FA(])p Fz(b)2133 4208 y FF(since)i Fz(c)p FA([)p Fz(s)2464 4223 y Fr(l)2491 4208 y FA(])j Fz(<)g(c)p FA([)p Fz(s)2744 4222 y Fr(r)2782 4208 y FA(])p FF(,)300 4431 y(thus)d Fz(c)p FA([)p Fz(s)606 4446 y Fr(l)632 4431 y Fz(s)p FA(])j Fz(<)g(c)p FA([)p Fz(s)928 4445 y Fr(r)966 4431 y Fz(s)p FA(])p FF(.)p 3534 4431 4 62 v 3538 4373 55 4 v 3538 4431 V 3592 4431 4 62 v 300 4675 a Fp(Proposition)h(3.8.)44 b Fw(Let)28 b Fz(s)33 b FD(2)g Fm(S)n FA(\()p FD(T)17 b FA(\))p Fw(,)28 b(and)g(let)g FB(T)p FA(\()p Fz(l)r FA(\))f Fw(and)h FB(T)p FA(\()p Fz(r)s FA(\))f Fw(be)g(two)h(transition)h(triples)g(in)e FD(T)22 b Fw(.)41 b(If)300 4844 y Fz(l)27 b(<)e(r)g Fw(and)e Fz(c)p FA([)p Fz(s)p FA(])j Fz(<)f FA(1)p Fw(,)e(then)g Fz(c)p FA([)p Fz(s)p FB(T)p FA(\()p Fz(l)r FA(\)])k Fz(<)e(c)p FA([)p Fz(s)p FB(T)p FA(\()p Fz(r)s FA(\)])p Fw(.)1905 5349 y FF(27)p eop %%Page: 28 33 28 32 bop 300 391 a Fw(Pr)n(oof.)46 b FF(First,)22 b(if)h Fz(l)28 b FA(=)d(0)p FF(,)d(then)g Fz(c)p FA([)p Fz(s)p FB(T)p FA(\()p Fz(l)r FA(\)])k(=)f(0)h Fz(<)f(c)p FA([)p Fz(s)p FB(T)p FA(\()p Fz(r)s FA(\)])p FF(,)e(since)g Fz(r)k(>)e(l)j FA(=)c(0)p FF(.)583 561 y(Let)e Fz(c)k FA(=)e Fz(c)p FA([)p Fz(s)p FA(])p FF(.)29 b(For)22 b(the)f(case)i Fz(l)k(>)e FA(0)p FF(,)e(we)f(have:)1935 783 y Fz(r)g FD(\000)e Fz(l)293 b(>)25 b FA(0)1935 977 y(1)20 b FD(\000)g Fz(c)279 b(>)25 b FA(0)1506 1172 y Fz(c)p FA(\(1)c FD(\000)f Fz(r)s FA(\))25 b Fz(<)e(c)p FA(\(1)e FD(\000)f Fz(l)r FA(\))179 b Fz(<)25 b FA(1)1349 1366 y(1)c FD(\000)f Fz(c)p FA(\(1)h FD(\000)f Fz(r)s FA(\))25 b Fz(>)e FA(1)d FD(\000)g Fz(c)p FA(\(1)i FD(\000)d Fz(l)r FA(\))k Fz(>)i FA(0)300 1589 y FF(So)1308 1750 y FA(\(1)c FD(\000)f Fz(c)p FA(\)\()p Fz(r)k FD(\000)c Fz(l)r FA(\))p 1039 1790 1061 4 v 1039 1873 a(\(1)h FD(\000)f Fz(c)p FA(\(1)h FD(\000)f Fz(r)s FA(\)\)\(1)h FD(\000)f Fz(c)p FA(\(1)h FD(\000)f Fz(l)r FA(\)\))2134 1811 y Fz(>)25 b FA(0)1858 1992 y Fz(l)p 1647 2033 453 4 v 1647 2116 a FA(1)20 b FD(\000)g Fz(c)p FA(\(1)i FD(\000)e Fz(l)r FA(\))2134 2054 y Fz(<)2452 1992 y(r)p 2240 2033 467 4 v 2240 2116 a FA(1)h FD(\000)f Fz(c)p FA(\(1)h FD(\000)f Fz(r)s FA(\))1497 2302 y Fz(l)i FA(+)1711 2240 y(\(1)f FD(\000)f Fz(l)r FA(\))p Fz(cl)p 1647 2281 453 4 v 1647 2364 a FA(1)g FD(\000)g Fz(c)p FA(\(1)i FD(\000)e Fz(l)r FA(\))1497 2423 y Fq(|)p 1538 2423 225 11 v 225 w({z)p 1845 2423 V 225 w(})1662 2513 y Fs(=)p Fr(c)p Fs([)p Fr(s)p Fd(T)p Fs(\()p Fr(l)q Fs(\)])2134 2302 y Fz(<)25 b(r)e FA(+)2451 2240 y(\(1)e FD(\000)f Fz(r)s FA(\))p Fz(cr)p 2395 2281 467 4 v 2395 2364 a FA(1)g FD(\000)g Fz(c)p FA(\(1)i FD(\000)d Fz(r)s FA(\))2230 2423 y Fq(|)p 2271 2423 239 11 v 239 w({z)p 2592 2423 V 239 w(})2404 2513 y Fs(=)p Fr(c)p Fs([)p Fr(s)p Fd(T)p Fs(\()p Fr(r)r Fs(\)])300 2668 y FF(Ther)n(efor)n(e,)i Fz(c)p FA([)p Fz(s)p FB(T)p FA(\()p Fz(l)r FA(\)])26 b Fz(<)f(c)p FA([)p Fz(s)p FB(T)p FA(\()p Fz(r)s FA(\)])p FF(.)p 3534 2668 4 62 v 3538 2610 55 4 v 3538 2668 V 3592 2668 4 62 v 300 2906 a Fp(Proposition)c (3.9.)41 b Fw(If)23 b Fz(t;)15 b(s)25 b FD(2)g Fm(S)o FA(\()p FD(T)17 b FA(\))p Fw(,)22 b Fz(c)p FA([)p Fz(t)p FA(])k Fz(<)f FA(1)p Fw(,)d Fz(c)p FA([)p Fz(s)p FA(])k Fz(<)f FA(1)p Fw(,)e(and)f Fz(l)27 b(<)e(r)s Fw(,)d(then)h Fz(c)p FA([)p Fz(t)p FB(T)p FA(\()p Fz(l)r FA(\))p Fz(s)p FA(])j Fz(<)f(c)p FA([)p Fz(t)p FB(T)p FA(\()p Fz(r)s FA(\))p Fz(s)p FA(])p Fw(.)300 3143 y(Pr)n(oof.)46 b FF(By)25 b(Pr)n(oposition)f(3.7,)j Fz(c)p FA([)p FB(T)p FA(\()p Fz(l)r FA(\))p Fz(s)p FA(])k Fz(<)f(c)p FA([)p FB(T)p FA(\()p Fz(r)s FA(\))p Fz(s)p FA(])p FF(.)37 b(Thus)24 b(it)i(follows)f(fr)n(om)h(Pr)n(oposition)e(3.8)300 3313 y(that)e Fz(c)p FA([)p Fz(t)p FB(T)p FA(\()p Fz(l)r FA(\))p Fz(s)p FA(])k Fz(<)f(c)p FA([)p Fz(t)p FB(T)p FA(\()p Fz(r)s FA(\))p Fz(s)p FA(])p FF(.)p 3534 3313 V 3538 3255 55 4 v 3538 3313 V 3592 3313 4 62 v 300 3551 a Fp(Lemma)34 b(3.10)i(\(Comparison)e(Lemma\).)48 b Fw(For)35 b(any)g Fz(s;)15 b(s)2278 3518 y Fn(0)2347 3551 y FD(2)47 b Fm(S)2511 3569 y Fs([)p Fr(a)p Fs(:)p Fr(b)p Fs(])2636 3551 y FA(\()p FD(T)23 b FA(\))35 b Fw(such)f(that)i Fz(\015)5 b FA([)p Fz(s)3317 3518 y Fn(0)3340 3551 y FA(\()p Fz(z)t FA(\)])48 b FD(\024)300 3720 y Fz(\015)5 b FA([)p Fz(s)p FA(\()p Fz(z)t FA(\)])24 b Fw(for)g(each)g Fz(z)29 b FD(2)c FA([)p Fz(a)h FA(:)f Fz(b)p FA(])p Fw(,)e(the)h(inequality)g Fz(c)p FA([)p Fz(s)1940 3687 y Fn(0)1963 3720 y FA(])i FD(\024)f Fz(c)p FA([)p Fz(s)p FA(])e Fw(holds.)300 3958 y(Pr)n(oof.)46 b FF(Applying)23 b(Pr)n(oposition)e(3.9)j(to)f(each)g (position)f Fz(z)30 b FD(2)c FA([)p Fz(a)g FA(:)h Fz(b)p FA(])c FF(wher)n(e)f Fz(\015)5 b FA([)p Fz(s)3049 3925 y Fn(0)3073 3958 y FA(\()p Fz(z)t FA(\)])27 b Fz(<)f(\015)5 b FA([)p Fz(s)p FA(\()p Fz(z)t FA(\)])300 4127 y FF(yields)22 b(this)g(r)n(esult.)p 3534 4127 V 3538 4069 55 4 v 3538 4127 V 3592 4127 4 62 v 300 4494 a Ft(3.3)119 b(Left-leaning)29 b(T)-11 b(ransition)29 b(Sets)300 4761 y FF(This)21 b(is)h(the)f (\002rst)f(in)i(a)g(series)e(of)i(special)g(cases)f(of)g(the)g(general) g(r)n(esult)g(as)h(stated)e(in)h(the)g(Main)300 4931 y(Theor)n(em.)36 b(The)25 b(intuition)g(her)n(e)g(for)h(left-leaning)g (transition)g(sets)e(is)h(that)h(the)f(points)f(in)i FA(\000)3553 4945 y Fr(n)300 5100 y FF(ar)n(e)d(gr)n(ouped)e(together) -7 b(,)20 b(and)i(these)f(gr)n(oups)h(r)n(e\002ne)g(and)g(evolve)h(in)g (a)g(self-similar)h(fashion.)1905 5349 y(28)p eop %%Page: 29 34 29 33 bop 583 391 a FF(Let)21 b FA(0)k Fz(<)g(x)952 405 y Fs(1)1017 391 y Fz(<)g(x)1165 405 y Fs(2)1230 391 y Fz(<)g FD(\001)15 b(\001)g(\001)26 b Fz(<)f(x)1605 406 y Fr(d)1671 391 y Fz(<)g FA(1)p Fz(=)p FA(2)e FF(be)e(a)h(non-empty)-10 b(,)20 b(\002nite)h(set)f(of)i(r)n(eal)g(numbers,)300 561 y(and)1380 730 y FD(T)47 b FA(=)25 b FD(f)p FB(T)1680 744 y Fr(i)1734 730 y FA(=)g FB(T)p FA(\()p Fz(x)1979 744 y Fr(i)2007 730 y FA(\))f FD(j)f Fz(i)i FD(2)g FA([1)h(:)f Fz(d)p FA(])p FD(g)300 951 y FF(be)e(a)g(transition)f(set.)583 1120 y(For)g(each)h Fz(x)999 1134 y Fr(i)1027 1120 y FF(,)g(de\002ne)e(the)h(uniform)h(strategy)d(sequence)h FA(\()p FB(T)2682 1076 y Fr(j)2682 1148 y(i)2719 1120 y FA(\))2754 1134 y Fr(j)t Fn(2)p Fl(N)2911 1120 y FA(=)k Fz(U)10 b FA(\()p FB(T)3176 1134 y Fr(i)3205 1120 y FA(\))p FF(.)300 1433 y Fo(3.3.1)99 b(Implicit)23 b(Construction)j(of)f(the)g (Limit)f(Set)300 1666 y FF(While)i(we)e(may)i(not)e(be)h(able)h(to)f (describe)g(the)f(exact)h(shape)g(of)g(the)f(limit)j(set,)d(we)h(can)h (con-)300 1835 y(str)o(uct)f(sequences)f(of)i(appr)n(oximations)g (which)g(themselves)e(conver)n(ge)i(to)f(the)g(limit)i(set.)37 b(It)300 2005 y(is)26 b(easier)g(to)f(work)h(with)g(these)e(appr)n (oximations)i(than)g(with)g(the)g(actual)h FA(\000)2997 2019 y Fr(n)3044 2005 y FF(,)g(as)f(they)f(have)300 2174 y(r)n(elatively)e(straightforwar)n(d)f(de\002nitions)g(compar)n(ed)g (to)g(that)g(of)h FA(\000)2681 2188 y Fr(n)2728 2174 y FF(.)300 2414 y Fp(De\002nition)f(3.1)m(1.)42 b FF(Let)21 b Fz(s)k FD(2)g Fm(S)o FA(\()p FD(T)17 b FA(\))p FF(.)28 b(De\002ne)22 b(the)g(sets)f Fz(L)p FA(\()p Fz(s)p FA(\))i FF(and)f Fz(I)7 b FA(\()p Fz(s)p FA(\))23 b FF(by)1227 2670 y Fz(L)p FA(\()p Fz(s)p FA(\))j(=)1524 2569 y Fq(n)1623 2670 y FA(lim)1584 2725 y Fr(m)p Fn(!1)1803 2670 y Fz(c)p FA([)p FB(T)1929 2633 y Fr(m)1929 2693 y(i)1996 2670 y Fz(s)p FA(])d FD(j)g Fz(i)i FD(2)g FA([1)h(:)g Fz(d)p FA(])2496 2569 y Fq(o)3426 2670 y FF(\(3.1\))1242 2886 y Fz(I)7 b FA(\()p Fz(s)p FA(\))26 b(=)1524 2785 y Fq(h)1605 2886 y FA(lim)1567 2940 y Fr(m)p Fn(!1)1786 2886 y Fz(c)p FA([)p FB(T)1912 2848 y Fr(m)1912 2908 y Fs(1)1979 2886 y Fz(s)p FA(])p Fz(;)53 b FA(lim)2087 2940 y Fr(m)p Fn(!1)2306 2886 y Fz(c)p FA([)p FB(T)2432 2848 y Fr(m)2432 2909 y(d)2499 2886 y Fz(s)p FA(])2567 2785 y Fq(i)2647 2886 y Fz(:)754 b FF(\(3.2\))583 3142 y(Now)22 b(de\002ne)f(the)h(sets)f Fz(L)1464 3156 y Fr(n)1534 3142 y FF(and)h Fz(I)1750 3156 y Fr(n)1820 3142 y FF(in)h(terms)f(of)g Fz(L)p FA(\()p Fz(s)p FA(\))h FF(and)g Fz(I)7 b FA(\()p Fz(s)p FA(\))23 b FF(by)1547 3398 y Fz(L)1609 3412 y Fr(n)1681 3398 y FA(=)1903 3312 y Fq([)1777 3513 y Fr(s)p Fn(2)p Fl(S)1899 3527 y Fk([1:)p Fj(n)p Fk(])2019 3513 y Fs(\()p Fn(T)17 b Fs(\))2145 3398 y Fz(L)p FA(\()p Fz(s)p FA(\))1106 b FF(\(3.3\))1569 3701 y Fz(I)1609 3715 y Fr(n)1681 3701 y FA(=)1903 3614 y Fq([)1777 3816 y Fr(s)p Fn(2)p Fl(S)1899 3830 y Fk([1:)p Fj(n)p Fk(])2019 3816 y Fs(\()p Fn(T)17 b Fs(\))2145 3701 y Fz(I)7 b FA(\()p Fz(s)p FA(\))23 b Fz(:)1073 b FF(\(3.4\))300 4013 y(Note)21 b(that)1768 4182 y Fz(L)1830 4196 y Fr(n)1902 4182 y FD(\022)k Fz(I)2038 4196 y Fr(n)2107 4182 y Fz(:)1294 b FF(\(3.5\))583 4423 y(The)35 b(sets)e Fz(L)1018 4437 y Fr(n)1100 4423 y FF(and)i Fz(I)1329 4437 y Fr(n)1410 4423 y FF(r)n(epr)n(esent,)h(r)n (espectively)-10 b(,)37 b(the)d(\223lower)7 b(\224)35 b(and)f(\223upper)7 b(\224)35 b(ap-)300 4592 y(pr)n(oximations)30 b(of)g(the)f(limit)i(set,)g(given)e(information)i(about)f(the)f (strategies)f Fm(S)3165 4610 y Fs([1:)p Fr(n)p Fs(])3300 4592 y FA(\()p FD(T)23 b FA(\))p FF(.)50 b(In)300 4761 y(particular)-7 b(,)35 b(the)c(set)f Fz(L)p FA(\()p Fz(s)p FA(\))i FF(contains)g(a)g(\002nite)f(number)g(of)h(isolated)f(points,)h (two)f(of)h(which)300 4931 y(ar)n(e)k(the)f(minimum)i(and)e(maximum)i (value)g(of)e Fz(c)p FA([)p Fz(s)2177 4898 y Fn(0)2201 4931 y Fz(s)p FA(])g FF(as)h(computed)e(over)h(all)i(strategies)300 5100 y Fz(s)343 5067 y Fn(0)392 5100 y FD(2)26 b Fm(S)o FA(\()p FD(T)17 b FA(\))p FF(,)23 b(while)h(the)e(set)g Fz(I)7 b FA(\()p Fz(s)p FA(\))24 b FF(is)f(the)g(interval)h(between)d (said)j(minimum)h(and)e(maximum.)1905 5349 y(29)p eop %%Page: 30 35 30 34 bop 300 391 a FF(So)22 b(intuitively)-10 b(,)24 b(the)e(limit)i(set)e(should)g(\223contain\224)j Fz(L)2146 405 y Fr(n)2215 391 y FF(for)e(all)i Fz(n)p FF(,)d(while)i(at)f(the)f (same)h(time)g(be)300 561 y(\223contained\224)d(by)g(all)h Fz(I)1086 575 y Fr(n)1153 561 y FF(for)f(all)h Fz(n)p FF(.)27 b(Mor)n(eover)-7 b(,)19 b(it)h(will)h(be)f(shown)e(that,)i(for) g(suf)n(\002ciently)g(lar)n(ge)300 730 y Fz(n)p FF(,)25 b(the)f(sets)g Fz(L)792 744 y Fr(n)838 730 y FF(,)i FA(\000)944 744 y Fr(n)1016 730 y FF(and)f Fz(I)1235 744 y Fr(n)1307 730 y FF(ar)n(e)g(all)i(quite)d(close)h(to)f(each)i(other)-7 b(,)24 b(with)h(distances)f(measur)n(ed)300 899 y(with)h(the)f (Hausdorf)n(f)h(metric.)35 b(Hence)24 b(the)g(limiting)i(natur)n(e)f (of)g Fz(L)2644 913 y Fr(n)2716 899 y FF(and)g Fz(I)2935 913 y Fr(n)3007 899 y FF(should)f(give)h(us)300 1069 y(the)d(desir)n(ed)f(r)n(esult,)h(namely)h FA(\000)1421 1083 y Fr(n)1490 1069 y FF(conver)n(ges)f(to)g(this)g(elusive)g(limit)i (set)e FA(\003)p FF(.)583 1238 y(In)32 b(or)n(der)f(to)h(establish)g (that)g(the)f(sets)f Fz(L)2038 1252 y Fr(n)2117 1238 y FF(and)j Fz(I)2344 1252 y Fr(n)2423 1238 y FF(do)e(indeed)g(conver)n (ge,)i(we)f(will)300 1408 y(pr)n(ove)d(that)g(the)f(sequence)g Fz(L)1365 1422 y Fr(n)1441 1408 y FF(is)i(monotonically)g(incr)n (easing,)h(while)e(the)g(sequence)f Fz(I)3459 1422 y Fr(n)3535 1408 y FF(is)300 1577 y(monotonically)23 b(decr)n(easing.)k (This)c(will)g(imply)h(that)e Fz(L)2263 1591 y Fr(n)2335 1577 y FD(!)2451 1509 y Fq(S)2527 1604 y Fr(n)2589 1577 y Fz(L)2651 1591 y Fr(n)2698 1577 y FF(,)g(and)h Fz(I)2960 1591 y Fr(n)3032 1577 y FD(!)3148 1509 y Fq(T)3224 1604 y Fr(n)3286 1577 y Fz(I)3326 1591 y Fr(n)3373 1577 y FF(.)300 1818 y Fp(Proposition)f(3.12.)42 b Fz(L)1093 1832 y Fr(n)1165 1818 y FD(\022)25 b Fz(L)1323 1832 y Fr(n)p Fs(+1)1460 1818 y Fw(.)300 2061 y(Pr)n(oof.)46 b FF(First,)31 b(if)g Fz(s)929 2028 y Fn(0)991 2061 y FD(2)39 b Fm(S)1147 2080 y Fs([1:)p Fr(n)p Fs(+)o(1])1373 2061 y FA(\()p FD(T)23 b FA(\))p FF(,)32 b(then)d Fz(s)1823 2028 y Fn(0)1885 2061 y FA(=)38 b FB(T)2056 2075 y Fr(i)2085 2061 y Fz(s)29 b FF(for)h(some)f FB(T)2607 2075 y Fr(i)2675 2061 y FD(2)38 b(T)53 b FF(and)30 b Fz(s)39 b FD(2)f Fm(S)3299 2080 y Fs([)o(1:)p Fr(n)p Fs(])3434 2061 y FA(\()p FD(T)23 b FA(\))p FF(.)300 2231 y(Hence,)e(we)i(begin)f(with)h (the)e(pr)n(oof)i(of)1553 2488 y Fz(L)p FA(\()p Fz(s)p FA(\))i FD(\022)1903 2401 y Fq([)1849 2603 y Fr(j)t Fn(2)p Fs([1:)p Fr(d)p Fs(])2074 2488 y Fz(L)p FA(\()p FB(T)2233 2502 y Fr(j)2270 2488 y Fz(s)p FA(\))1078 b FF(\(3.6\))300 2791 y(with)23 b(the)e(observation)h(that)1030 3048 y Fz(L)p FA(\()p Fz(s)p FA(\))j(=)1326 2947 y Fq(n)1425 3048 y FA(lim)1387 3102 y Fr(m)p Fn(!1)1605 3048 y Fz(c)p FA([)p FB(T)1731 3010 y Fr(m)1731 3070 y(i)1798 3048 y Fz(s)p FA(])e FD(j)g Fz(i)i FD(2)g FA([1)h(:)g Fz(d)p FA(])2298 2947 y Fq(o)1230 3263 y FA(=)1326 3162 y Fq(n)1425 3263 y FA(lim)1387 3318 y Fr(m)p Fn(!1)1605 3263 y Fz(c)p FA([)p FB(T)1731 3225 y Fr(m)1731 3286 y(i)1798 3263 y FB(T)1860 3277 y Fr(i)1888 3263 y Fz(s)p FA(])d FD(j)g Fz(i)i FD(2)g FA([1)h(:)g Fz(d)p FA(])2388 3162 y Fq(o)1230 3478 y FD(\022)1326 3378 y Fq(n)1425 3478 y FA(lim)1387 3533 y Fr(m)p Fn(!1)1605 3478 y Fz(c)p FA([)p FB(T)1731 3441 y Fr(m)1731 3501 y(i)1798 3478 y FB(T)1860 3492 y Fr(j)1897 3478 y Fz(s)p FA(])c FD(j)h Fz(i)j FD(2)f FA([1)h(:)f Fz(d)p FA(])p Fz(;)15 b(j)32 b FD(2)25 b FA([1)h(:)f Fz(d)p FA(])2809 3378 y Fq(o)1230 3694 y FA(=)1380 3607 y Fq([)1326 3809 y Fr(j)t Fn(2)p Fs([1:)p Fr(d)p Fs(])1551 3593 y Fq(n)1650 3694 y FA(lim)1611 3748 y Fr(m)p Fn(!1)1830 3694 y Fz(c)p FA([)p FB(T)1956 3656 y Fr(m)1956 3716 y(i)2023 3694 y FB(T)2085 3708 y Fr(j)2122 3694 y Fz(s)p FA(])d FD(j)h Fz(i)j FD(2)f FA([1)h(:)f Fz(d)p FA(])2621 3593 y Fq(o)1230 3986 y FA(=)1380 3900 y Fq([)1326 4101 y Fr(j)t Fn(2)p Fs([1:)p Fr(d)p Fs(])1551 3986 y Fz(L)p FA(\()p FB(T)1710 4000 y Fr(j)1746 3986 y Fz(s)p FA(\))e Fz(:)583 4286 y FF(Now)-8 b(,)22 b(since)404 4543 y Fz(L)466 4557 y Fr(n)538 4543 y FA(=)760 4457 y Fq([)634 4658 y Fr(s)p Fn(2)p Fl(S)757 4672 y Fk([)o(1:)p Fj(n)p Fk(])876 4658 y Fs(\()p Fn(T)c Fs(\))1003 4543 y Fz(L)p FA(\()p Fz(s)p FA(\))25 b FD(\022)1425 4457 y Fq([)1299 4658 y Fr(s)p Fn(2)p Fl(S)1422 4672 y Fk([)o(1:)p Fj(n)p Fk(])1541 4658 y Fs(\()p Fn(T)18 b Fs(\))1722 4457 y Fq([)1667 4658 y Fr(j)t Fn(2)p Fs([1:)p Fr(d)p Fs(])1892 4543 y Fz(L)p FA(\()p FB(T)2051 4557 y Fr(j)2088 4543 y Fz(s)p FA(\))25 b(=)2463 4457 y Fq([)2287 4658 y Fr(s)2320 4639 y Fi(0)2342 4658 y Fn(2)p Fl(S)2432 4672 y Fk([1:)p Fj(n)p Fk(+)o(1])2629 4658 y Fs(\()p Fn(T)17 b Fs(\))2755 4543 y Fz(L)p FA(\()p Fz(s)2895 4506 y Fn(0)2918 4543 y FA(\))26 b(=)f Fz(L)3137 4557 y Fr(n)p Fs(+1)3296 4543 y Fz(;)105 b FF(\(3.7\))300 4857 y(we)22 b(can)i(infer)e(the)g(r)n(esult)g(that)h Fz(L)1457 4871 y Fr(n)1529 4857 y FD(\022)i Fz(L)1687 4871 y Fr(n)p Fs(+1)1824 4857 y FF(.)p 3534 4857 4 62 v 3538 4798 55 4 v 3538 4857 V 3592 4857 4 62 v 300 5100 a Fp(Proposition)d(3.13.)42 b Fz(I)1071 5114 y Fr(n)1143 5100 y FD(\023)25 b Fz(I)1279 5114 y Fr(n)p Fs(+1)1416 5100 y Fw(.)1905 5349 y FF(30)p eop %%Page: 31 36 31 35 bop 300 391 a Fw(Pr)n(oof.)46 b FF(Again,)23 b(we)f(will)i(begin) e(with)h(the)f(pr)n(oof)g(of)1567 652 y Fz(I)7 b FA(\()p Fz(s)p FA(\))26 b FD(\023)1903 566 y Fq([)1849 767 y Fr(j)t Fn(2)p Fs([1:)p Fr(d)p Fs(])2074 652 y Fz(I)7 b FA(\()p FB(T)2218 666 y Fr(j)2255 652 y Fz(s)p FA(\))1093 b FF(\(3.8\))300 959 y(by)23 b(observing)e(that)423 1219 y Fz(I)7 b FA(\()p Fz(s)p FA(\))26 b(=)705 1118 y Fq(h)786 1219 y FA(lim)748 1274 y Fr(m)p Fn(!1)966 1219 y Fz(c)p FA([)p FB(T)1092 1182 y Fr(m)1092 1242 y Fs(1)1160 1219 y Fz(s)p FA(])p Fz(;)53 b FA(lim)1268 1274 y Fr(m)p Fn(!1)1487 1219 y Fz(c)p FA([)p FB(T)1613 1182 y Fr(m)1613 1242 y(d)1680 1219 y Fz(s)p FA(])1748 1118 y Fq(i)609 1435 y FA(=)705 1334 y Fq(h)786 1435 y FA(lim)748 1489 y Fr(m)p Fn(!1)966 1435 y Fz(c)p FA([)p FB(T)1092 1397 y Fr(m)1092 1457 y Fs(1)1160 1435 y FB(T)1222 1449 y Fs(1)1261 1435 y Fz(s)p FA(])p Fz(;)h FA(lim)1369 1489 y Fr(m)p Fn(!1)1588 1435 y Fz(c)p FA([)p FB(T)1714 1397 y Fr(m)1714 1458 y(d)1781 1435 y FB(T)1843 1450 y Fr(d)1883 1435 y Fz(s)p FA(])1951 1334 y Fq(i)609 1650 y FD(\023)759 1564 y Fq([)705 1765 y Fr(j)t Fn(2)p Fs([1:)p Fr(d)p Fs(])930 1549 y Fq(h)1011 1650 y FA(lim)973 1705 y Fr(m)p Fn(!1)1191 1650 y Fz(c)p FA([)p FB(T)1317 1612 y Fr(m)1317 1672 y Fs(1)1385 1650 y FB(T)1447 1664 y Fr(j)1483 1650 y Fz(s)p FA(])p Fz(;)g FA(lim)1591 1705 y Fr(m)p Fn(!1)1810 1650 y Fz(c)p FA([)p FB(T)1936 1612 y Fr(m)1936 1673 y(d)2003 1650 y FB(T)2065 1664 y Fr(j)2101 1650 y Fz(s)p FA(])2169 1549 y Fq(i)2318 1650 y FF(\(by)23 b(the)f(Comparison)g (Lemma\))609 1942 y FA(=)759 1856 y Fq([)705 2057 y Fr(j)t Fn(2)p Fs([1:)p Fr(d)p Fs(])930 1942 y Fz(I)7 b FA(\()p FB(T)1074 1956 y Fr(j)1111 1942 y Fz(s)p FA(\))22 b Fz(:)583 2249 y FF(Ther)n(efor)n(e,)g(since)448 2510 y Fz(I)488 2524 y Fr(n)561 2510 y FA(=)783 2423 y Fq([)657 2625 y Fr(s)p Fn(2)p Fl(S)779 2639 y Fk([1:)p Fj(n)p Fk(])899 2625 y Fs(\()p Fn(T)17 b Fs(\))1025 2510 y Fz(I)7 b FA(\()p Fz(s)p FA(\))25 b FD(\023)1433 2423 y Fq([)1306 2625 y Fr(s)p Fn(2)p Fl(S)1429 2639 y Fk([1:)p Fj(n)p Fk(])1549 2625 y Fs(\()p Fn(T)17 b Fs(\))1729 2423 y Fq([)1675 2625 y Fr(j)t Fn(2)p Fs([1:)p Fr(d)p Fs(])1900 2510 y Fz(I)7 b FA(\()p FB(T)2044 2524 y Fr(j)2081 2510 y Fz(s)p FA(\))25 b(=)2456 2423 y Fq([)2280 2625 y Fr(s)2313 2606 y Fi(0)2335 2625 y Fn(2)p Fl(S)2425 2639 y Fk([)o(1:)p Fj(n)p Fk(+1])2621 2625 y Fs(\()p Fn(T)18 b Fs(\))2747 2510 y Fz(I)7 b FA(\()p Fz(s)2872 2472 y Fn(0)2896 2510 y FA(\))25 b(=)g Fz(I)3092 2524 y Fr(n)p Fs(+1)3252 2510 y Fz(;)149 b FF(\(3.9\))300 2826 y(we)22 b(can)i(conclude)e(that)g Fz(I)1210 2840 y Fr(n)1282 2826 y FD(\023)j Fz(I)1418 2840 y Fr(n)p Fs(+1)1556 2826 y FF(.)p 3534 2826 4 62 v 3538 2768 55 4 v 3538 2826 V 3592 2826 4 62 v 583 3071 a(W)-5 b(ith)21 b(the)e(conver)n(gence)g(of)h Fz(L)1630 3085 y Fr(n)1697 3071 y FF(and)g Fz(I)1911 3085 y Fr(n)1978 3071 y FF(pr)n(oven,)g(we)f(pr)n(oceed)g(to)h(show)f(that)g(the)h(sets) 300 3240 y Fz(L)362 3254 y Fr(n)409 3240 y FF(,)i FA(\000)511 3254 y Fr(n)558 3240 y FF(,)h(and)f Fz(I)820 3254 y Fr(n)890 3240 y FF(ar)n(e)h(close)f(for)h(suf)n(\002ciently)f(lar)n(ge)h(values) g(of)g Fz(n)p FF(.)300 3484 y Fp(Lemma)29 b(3.14.)47 b Fw(Given)29 b(an)h(arbitrary)i Fz(\017)37 b(>)h FA(0)p Fw(,)31 b(ther)n(e)g(is)f(an)f Fz(N)2370 3498 y Fs(0)2447 3484 y FD(2)38 b Fm(N)k Fw(such)29 b(that)i(for)f(all)h Fz(n)37 b(>)g(N)3538 3498 y Fs(0)3577 3484 y Fw(,)300 3653 y Fz(B)369 3667 y Fr(\017)402 3653 y FA(\(\000)494 3667 y Fr(n)541 3653 y FA(\))25 b FD(\023)g Fz(I)737 3667 y Fr(n)784 3653 y Fw(.)300 3897 y(Pr)n(oof.)46 b FF(As)32 b FA(lim)831 3911 y Fr(n)p Fn(!1)1034 3897 y Fz(d)p FA([)p FB(T)1168 3864 y Fr(n)1168 3925 y(d)1216 3897 y FA(])43 b(=)f(0)p FF(,)35 b(let)d(us)f(de\002ne)g Fz(N)2113 3911 y Fs(0)2185 3897 y FF(to)g(be)h(such)g(that)g Fz(d)p FA([)p FB(T)2965 3864 y Fr(n)2965 3925 y(d)3012 3897 y FA(])43 b Fz(<)g(\017=)p FA(2)33 b FF(for)f(all)300 4067 y Fz(n)25 b(>)g(N)549 4081 y Fs(0)588 4067 y FF(.)j(Now)-8 b(,)21 b(let)i Fz(s)h FD(2)h Fm(S)1210 4085 y Fs([)o(1:)p Fr(n)p Fs(])1345 4067 y FA(\()p FD(T)e FA(\))p FF(,)g(wher)n(e)f Fz(n)i(>)h(N)2055 4081 y Fs(0)2095 4067 y FF(.)j(Then,)21 b(for)i(all)h Fz(i)h FD(2)g FA([1)h(:)g Fz(d)p FA(])d FF(and)g Fz(m)i FD(2)g Fm(N)6 b FF(,)528 4328 y Fz(c)p FA([)p FB(T)654 4290 y Fr(m)654 4350 y(i)721 4328 y Fz(s)p FA(])21 b FD(\000)e Fz(c)p FA([)p Fz(s)p FA(])26 b(=)f Fz(a)p FA([)p FB(T)1289 4290 y Fr(m)1289 4350 y(i)1356 4328 y FA(]\(1)c FD(\000)f Fz(c)p FA([)p FB(T)1699 4290 y Fr(m)1699 4350 y(i)1766 4328 y FA(])p Fz(b)p FA([)p Fz(s)p FA(]\))1959 4286 y Fn(\000)p Fs(1)2053 4328 y Fz(c)p FA([)p FB(T)2179 4290 y Fr(m)2179 4350 y(i)2246 4328 y FA(])p Fz(d)p FA([)p Fz(s)p FA(])1058 4522 y FD(\024)25 b Fz(d)p FA([)p Fz(s)p FA(])208 b Fe(*)24 b Fz(a)p FA([)p FB(T)1722 4484 y Fr(m)1722 4544 y(i)1789 4522 y FA(]\(1)d FD(\000)f Fz(c)p FA([)p FB(T)2132 4484 y Fr(m)2132 4544 y(i)2199 4522 y FA(])p Fz(b)p FA([)p Fz(s)p FA(]\))2392 4480 y Fn(\000)p Fs(1)2486 4522 y Fz(c)p FA([)p FB(T)2612 4484 y Fr(m)2612 4544 y(i)2680 4522 y FA(])25 b FD(\024)g FA(1)1058 4716 y FD(\024)g Fz(d)p FA([)p FB(T)1288 4679 y Fr(n)1288 4739 y(d)1336 4716 y FA(])182 b FF(by)22 b(the)g(Comparison)h(Lemma)459 b Fz(<)25 b(\017=)p FA(2)e Fz(:)1905 5349 y FF(31)p eop %%Page: 32 37 32 36 bop 583 391 a FF(Ther)n(efor)n(e,)20 b FA(lim)1141 405 y Fr(m)p Fn(!1)1364 391 y Fz(c)p FA([)p FB(T)1490 358 y Fr(m)1490 416 y Fs(1)1557 391 y Fz(s)1600 405 y Fr(n)1647 391 y FA(])14 b FD(\000)g Fz(c)p FA([)p Fz(s)1878 405 y Fr(n)1924 391 y FA(])26 b FD(\024)f Fz(\017=)p FA(2)c FF(and)g FA(lim)2520 405 y Fr(m)p Fn(!1)2743 391 y Fz(c)p FA([)p FB(T)2869 358 y Fr(m)2869 419 y(d)2936 391 y Fz(s)2979 405 y Fr(n)3026 391 y FA(])14 b FD(\000)g Fz(c)p FA([)p Fz(s)3257 405 y Fr(n)3303 391 y FA(])26 b FD(\024)e Fz(\017=)p FA(2)p FF(,)300 561 y(leading)f(us)f(to)g (conclude)g(that)1266 821 y Fz(I)7 b FA(\()p Fz(s)p FA(\))26 b(=)1548 720 y Fq(h)1630 821 y FA(lim)1591 876 y Fr(m)p Fn(!1)1810 821 y Fz(c)p FA([)p FB(T)1936 784 y Fr(m)1936 844 y Fs(1)2003 821 y Fz(s)p FA(])p Fz(;)54 b FA(lim)2111 876 y Fr(m)p Fn(!1)2330 821 y Fz(c)p FA([)p FB(T)2456 784 y Fr(m)2456 845 y(d)2523 821 y Fz(s)p FA(])2591 720 y Fq(i)1452 1037 y FD(\022)25 b Fz(B)1617 1051 y Fr(\017)1650 1037 y FA(\()p Fz(c)p FA([)p Fz(s)p FA(]\))300 1322 y FF(for)e(all)h Fz(s)g FD(2)h Fm(S)770 1341 y Fs([)o(1:)p Fr(n)p Fs(])905 1322 y FA(\()p FD(T)e FA(\))p FF(,)g(and)f(hence)1340 1608 y Fz(I)1380 1622 y Fr(n)1452 1608 y FA(=)1674 1522 y Fq([)1548 1723 y Fr(s)p Fn(2)p Fl(S)1671 1737 y Fk([)o(1:)p Fj(n)p Fk(])1790 1723 y Fs(\()p Fn(T)c Fs(\))1916 1608 y Fz(I)7 b FA(\()p Fz(s)p FA(\))1452 1911 y FD(\022)1674 1824 y Fq([)1548 2026 y Fr(s)p Fn(2)p Fl(S)1671 2040 y Fk([)o(1:)p Fj(n)p Fk(])1790 2026 y Fs(\()p Fn(T)18 b Fs(\))1916 1911 y Fz(B)1985 1925 y Fr(\017)2018 1911 y FA(\()p Fz(c)p FA([)p Fz(s)p FA(]\))26 b(=)f Fz(B)2411 1925 y Fr(\017)2444 1911 y FA(\(\000)2536 1925 y Fr(n)2583 1911 y FA(\))300 2228 y FF(for)e(all)h Fz(n)g(>)h(N)808 2242 y Fs(0)848 2228 y FF(.)p 3534 2397 4 62 v 3538 2339 55 4 v 3538 2397 V 3592 2397 4 62 v 300 2641 a Fp(Lemma)k(3.15.)47 b Fw(Given)29 b(an)h(arbitrary)i Fz(\017)37 b(>)h FA(0)p Fw(,)31 b(ther)n(e)g(is)f(an)f Fz(N)2370 2655 y Fs(1)2447 2641 y FD(2)38 b Fm(N)k Fw(such)29 b(that)i(for)f(all)h Fz(n)37 b(>)g(N)3538 2655 y Fs(1)3577 2641 y Fw(,)300 2810 y FA(\000)357 2824 y Fr(n)429 2810 y FD(\022)25 b Fz(B)594 2824 y Fr(\017)627 2810 y FA(\()p Fz(L)724 2824 y Fr(n)771 2810 y FA(\))p Fw(.)300 3054 y(Pr)n(oof.)46 b FF(As)32 b FA(lim)831 3068 y Fr(n)p Fn(!1)1034 3054 y Fz(d)p FA([)p FB(T)1168 3021 y Fr(n)1168 3082 y(d)1216 3054 y FA(])43 b(=)f(0)p FF(,)35 b(let)d(us)f(de\002ne)g Fz(N)2113 3068 y Fs(1)2185 3054 y FF(to)g(be)h(such)g(that)g Fz(d)p FA([)p FB(T)2965 3021 y Fr(n)2965 3082 y(d)3012 3054 y FA(])43 b Fz(<)g(\017=)p FA(2)33 b FF(for)f(all)300 3224 y Fz(n)25 b(>)h(N)550 3238 y Fs(1)589 3224 y FF(.)j(Now)-8 b(,)22 b(let)g Fz(s)k FD(2)f Fm(S)1214 3242 y Fs([)o(1:)p Fr(n)p Fs(])1349 3224 y FA(\()p FD(T)e FA(\))p FF(,)g(wher)n(e)f Fz(n)j(>)h(N)2061 3238 y Fs(1)2100 3224 y FF(.)j(Then,)22 b(by)h(the)f(r)n(easoning)g(employed)g(in)300 3393 y(the)g(pr)n(evious) g(pr)n(oof,)g Fz(c)p FA([)p FB(T)1215 3360 y Fr(m)1215 3419 y(i)1282 3393 y Fz(s)p FA(])e FD(\000)g Fz(c)p FA([)p Fz(s)p FA(])25 b Fz(<)g(\017=)p FA(2)f FF(for)e(all)j Fz(i)g FD(2)g FA([1)h(:)f Fz(d)p FA(])f FF(and)e Fz(m)j FD(2)g Fm(N)7 b FF(.)583 3563 y(Ther)n(efor)n(e,)33 b FA(lim)1154 3577 y Fr(m)p Fn(!1)1377 3563 y Fz(c)p FA([)p FB(T)1503 3530 y Fr(m)1503 3588 y(i)1570 3563 y Fz(s)1613 3577 y Fr(n)1659 3563 y FA(])28 b FD(\000)e Fz(c)p FA([)p Fz(s)1916 3577 y Fr(n)1963 3563 y FA(])42 b FD(\024)g Fz(\017=)p FA(2)32 b FF(for)f(all)i Fz(i)42 b FD(2)g FA([1)g(:)g Fz(d)p FA(])p FF(,)34 b(leading)e(us)f(to)300 3732 y(conclude)22 b(that)1885 3993 y Fz(\016)s FA(\()p FD(f)p Fz(c)p FA([)p Fz(s)p FA(])p FD(g)p Fz(;)15 b(L)p FA(\()p Fz(s)p FA(\)\))28 b Fz(<)d(\017)1060 4263 y(\016)1119 4080 y Fq(0)1119 4244 y(@)1324 4177 y([)1198 4378 y Fr(s)p Fn(2)p Fl(S)1321 4392 y Fk([)o(1:)p Fj(n)p Fk(])1440 4378 y Fs(\()p Fn(T)18 b Fs(\))1551 4263 y FD(f)p Fz(c)p FA([)p Fz(s)p FA(])p FD(g)p Fz(;)1941 4177 y Fq([)1813 4378 y Fr(s)p Fn(2)p Fl(S)1938 4392 y Fk([)o(1:)p Fj(n)p Fk(])2057 4378 y Fs(\()p Fn(T)g Fs(\))2183 4263 y Fz(L)p FA(\()p Fz(s)p FA(\))2358 4080 y Fq(1)2358 4244 y(A)2463 4263 y Fz(<)25 b(\017)2070 4566 y(\016)s FA(\(\000)2205 4580 y Fr(n)2253 4566 y Fz(;)15 b(L)2355 4580 y Fr(n)2403 4566 y FA(\))25 b Fz(<)g(\017)2334 4760 y FA(\000)2391 4774 y Fr(n)2463 4760 y FD(\022)g Fz(B)2628 4774 y Fr(\017)2661 4760 y FA(\()p Fz(L)2758 4774 y Fr(n)2805 4760 y FA(\))300 5021 y FF(for)e(all)h Fz(n)g(>)h(N)808 5035 y Fs(1)848 5021 y FF(.)p 3534 5021 V 3538 4963 55 4 v 3538 5021 V 3592 5021 4 62 v 1905 5349 a(32)p eop %%Page: 33 38 33 37 bop 583 391 a FF(Having)33 b(just)e(pr)n(oven)h(the)f(pr)n (oximity)h(of)g Fz(L)2176 405 y Fr(n)2223 391 y FF(,)i FA(\000)2337 405 y Fr(n)2416 391 y FF(and)f Fz(I)2643 405 y Fr(n)2722 391 y FF(to)e(each)i(other)-7 b(,)33 b(we)e(can)300 561 y(\002nally)37 b(identify)f(the)f(candidate)h(for)g (the)f(limit)j(set)d FA(\003)p FF(.)68 b(The)35 b(set)g(that)h(we)g (will)h(consider)300 730 y(is)d FA(\003)46 b(=)625 662 y Fq(T)701 757 y Fr(n)763 730 y Fz(I)803 744 y Fr(n)850 730 y FF(.)61 b(Befor)n(e)33 b(pr)n(oving)h(the)f(conver)n(gence)g(r)n (esult,)j(we)d(will)i(need)e(to)g(establish)300 899 y(that)27 b(ther)n(e)f(is)h(no)f(gap)h(between)f(the)g(limit)p 1851 820 247 4 v 1851 831 a Fq(S)1927 926 y Fr(n)1989 899 y Fz(L)2051 913 y Fr(n)2098 899 y FF(,)i(and)f(the)f(limit)2699 831 y Fq(T)2775 926 y Fr(n)2837 899 y Fz(I)2877 913 y Fr(n)2924 899 y FF(,)i(since)f(otherwise)300 1069 y(ther)n(e)20 b(may)i(be)f(mor)n(e)h(than)f(one)g(choice)g(for)h FA(\003)p FF(,)f(which)h(would)f(work)f(against)i(the)e(pr)n(oof)h(that)300 1238 y FA(\000)357 1252 y Fr(n)429 1238 y FD(!)k FA(\003)p FF(.)583 1408 y(Note)j(that)g(we)g(have)h(to)f(take)g(the)g(closur)n(e) g(of)2253 1339 y Fq(S)2328 1434 y Fr(n)2390 1408 y Fz(L)2452 1422 y Fr(n)2499 1408 y FF(,)i(because)e(while)3151 1339 y Fq(S)3227 1434 y Fr(n)3289 1408 y Fz(L)3351 1422 y Fr(n)3427 1408 y FF(con-)300 1577 y(tains)k(a)h(point)f FA(lim)975 1591 y Fr(m)p Fn(!1)1198 1577 y Fz(c)p FA([)p FB(T)1324 1544 y Fr(m)1324 1602 y(i)1391 1577 y Fz(s)p FA(])g FF(for)h(each)f(\002nite)g(strategy)f Fz(s)43 b FD(2)f Fm(S)o FA(\()p FD(T)17 b FA(\))33 b FF(and)f Fz(i)44 b FD(2)e FA([1)i(:)g Fz(d)p FA(])p FF(,)35 b(it)300 1746 y(fails)29 b(to)d(captur)n(e)i(monotone)d(strategy)h(sequences)f (whose)h(\223limiting\224)k(sequences)25 b(ar)n(e)j(\223ir)n(-)300 1916 y(rational\224,)d(in)g(the)e(sense)f(that)h(they)g(do)g(not)g (terminate)h(with)g(an)g(in\002nite)g(string)f(of)h FB(T)3344 1930 y Fr(i)3372 1916 y FF(.)31 b(This)300 2085 y(is)d(similar)h(to)e (the)g(case)h(of)f(the)g(Cantor)h(set,)f(wher)n(e)g(the)g(majority)h (of)g(the)f(points)f(in)i(the)f(set)300 2254 y(ar)n(e)k(not)f(the)f (endpoints)g(of)h(the)g(intervals)h(r)n(emoved)f(\(cf.)2420 2186 y Fq(S)2496 2281 y Fr(n)2558 2254 y Fz(L)2620 2268 y Fr(n)2667 2254 y FF(\),)i(but)f(rather)f(their)g(limit)300 2424 y(points)22 b(\(cf.)p 727 2345 V 727 2356 a Fq(S)803 2451 y Fr(n)865 2424 y Fz(L)927 2438 y Fr(n)974 2424 y FF(\),)h(which)h(happen)e(to)g(be)h(exactly)g(the)f(points)g(in)h (the)f(intervals)h(that)g(wer)n(e)300 2593 y(not)f(r)n(emoved)g(\(cf.) 988 2525 y Fq(T)1063 2620 y Fr(n)1126 2593 y Fz(I)1166 2607 y Fr(n)1212 2593 y FF(\).)300 2826 y Fp(Lemma)h(3.16.)p 854 2747 V 854 2758 a Fq(S)930 2853 y Fr(n)992 2826 y Fz(L)1054 2840 y Fr(n)1126 2826 y FA(=)1222 2758 y Fq(T)1298 2853 y Fr(n)1360 2826 y Fz(I)1400 2840 y Fr(n)300 3068 y Fw(Pr)n(oof.)46 b FF(First,)22 b(we)g(shall)i(show)p 1384 2989 V 1384 2999 a Fq(S)1460 3094 y Fr(n)1522 3068 y Fz(L)1584 3082 y Fr(n)1656 3068 y FD(\022)1752 2999 y Fq(T)1828 3094 y Fr(n)1890 3068 y Fz(I)1930 3082 y Fr(n)1977 3068 y FF(.)583 3237 y(Let)j Fz(z)38 b FD(2)p 915 3158 V 915 3169 a Fq(S)991 3264 y Fr(n)1053 3237 y Fz(L)1115 3251 y Fr(n)1161 3237 y FF(.)43 b(The)26 b(de\002nition)h(of)g(the)g(closur)n(e)g(of)g(a)h(set)f(implies)h(that) f(ther)n(e)f(is)i(a)300 3406 y(sequence)f(lying)i(strictly)g(in)1348 3338 y Fq(S)1424 3433 y Fr(n)1486 3406 y Fz(L)1548 3420 y Fr(n)1624 3406 y FF(that)f(conver)n(ges)g(to)g Fz(z)t FF(.)47 b(Thus,)30 b(we)e(may)h(\002nd)g(a)g(subse-)300 3576 y(quence)24 b FA(\()p Fz(z)688 3590 y Fr(i)717 3576 y FA(\))h FF(of)g(such)f(a)h(sequence,)e(wher)n(e)h Fz(z)1896 3590 y Fr(i)1954 3576 y FD(2)k Fz(L)2105 3590 y Fr(n)2148 3600 y Fj(i)2179 3576 y FF(,)d(and)f Fz(n)2460 3590 y Fr(i)2517 3576 y Fz(<)29 b(n)2672 3590 y Fr(j)2733 3576 y FF(for)24 b(all)j Fz(i)i(<)g(j)5 b FF(.)34 b(Suppose)300 3745 y Fz(z)47 b(=)-56 b FD(2)471 3677 y Fq(T)547 3772 y Fr(n)609 3745 y Fz(I)649 3759 y Fr(n)696 3745 y FF(.)39 b(Then)26 b(it)g(must)g(be)g(that)g Fz(z)47 b(=)-55 b FD(2)31 b Fz(I)1817 3759 y Fr(N)1911 3745 y FF(for)26 b(some)g Fz(N)10 b FF(.)39 b(By)25 b(Pr)n(oposition)g(3.13,)k Fz(I)3312 3759 y Fr(n)3391 3745 y FD(\023)i Fz(I)3533 3759 y Fr(m)300 3914 y FF(for)h(all)h Fz(n)42 b(<)h(m)p FF(,)34 b(and)e(hence)f Fz(z)57 b(=)-55 b FD(2)42 b Fz(I)1609 3928 y Fr(n)1688 3914 y FF(for)32 b(all)h Fz(n)42 b(>)h(N)10 b FF(.)56 b(Let)30 b Fz(\017)43 b FA(=)f Fz(\016)s FA(\()p FD(f)p Fz(z)t FD(g)p Fz(;)15 b(I)2986 3928 y Fr(N)3056 3914 y FA(\))43 b Fz(>)f FA(0)p FF(,)35 b(wher)n(e)300 4084 y Fz(\016)s FA(\()p FD(\001)p Fz(;)15 b FD(\001)p FA(\))28 b FF(is)e(as)f(de\002ned)f(in)i(\(2.27\))r(.)36 b(So)25 b Fz(\016)s FA(\()p Fz(z)t(;)15 b(I)1774 4098 y Fr(n)1823 4084 y FA(\))31 b FD(\025)f Fz(\016)s FA(\()p FD(f)p Fz(z)t FD(g)p Fz(;)15 b(I)2284 4098 y Fr(N)2354 4084 y FA(\))31 b(=)g Fz(\017)f(>)h FA(0)26 b FF(for)f(all)i Fz(n)j(>)h(N)10 b FF(.)36 b(Mor)n(e-)300 4253 y(over)-7 b(,)31 b(as)e Fz(L)703 4267 y Fr(n)788 4253 y FD(\032)38 b Fz(I)937 4267 y Fr(n)1014 4253 y FF(for)29 b(all)i Fz(n)p FF(,)g Fz(\016)s FA(\()p FD(f)p Fz(z)t FD(g)p Fz(;)15 b(L)1712 4267 y Fr(n)1761 4253 y FA(\))39 b FD(\025)e Fz(\017)30 b FF(for)f(all)i Fz(n)38 b(>)g(N)10 b FF(.)48 b(However)-7 b(,)30 b(since)g Fz(z)3336 4267 y Fr(i)3402 4253 y FD(!)38 b Fz(z)t FF(,)300 4423 y Fz(\016)s FA(\()p FD(f)p Fz(z)t FD(g)p Fz(;)15 b(L)616 4437 y Fr(n)666 4423 y FA(\))29 b FD(!)g FA(0)d FF(as)f Fz(n)k FD(!)g(1)p FF(.)34 b(This)25 b(is)g(a)g(contradiction.)34 b(Ther)n(efor)n(e)24 b(we)g(can)i(conclude)e(that)300 4592 y(if)f Fz(z)30 b FD(2)p 537 4513 V 537 4524 a Fq(S)613 4619 y Fr(n)675 4592 y Fz(L)737 4606 y Fr(n)784 4592 y FF(,)22 b(then)g Fz(z)29 b FD(2)1188 4524 y Fq(T)1264 4619 y Fr(n)1326 4592 y Fz(I)1366 4606 y Fr(n)1413 4592 y FF(.)583 4761 y(Now)-8 b(,)22 b(we)g(will)i(the)d(show)h(the)g(r)n(everse)f (inclusion)p 2384 4682 V 2384 4693 a Fq(S)2459 4788 y Fr(n)2521 4761 y Fz(L)2583 4775 y Fr(n)2656 4761 y FD(\023)2751 4693 y Fq(T)2827 4788 y Fr(n)2889 4761 y Fz(I)2929 4775 y Fr(n)2976 4761 y FF(.)583 4931 y(Let)30 b Fz(z)44 b FD(2)929 4862 y Fq(T)1005 4958 y Fr(n)1067 4931 y Fz(I)1107 4945 y Fr(n)1154 4931 y FF(.)51 b(Suppose)28 b Fz(z)54 b(=)-55 b FD(2)p 1788 4852 V 1788 4862 a Fq(S)1864 4958 y Fr(n)1926 4931 y Fz(L)1988 4945 y Fr(n)2035 4931 y FF(.)51 b(Then)30 b(ther)n(e)f(exists)g(an)i Fz(\017)40 b(>)f FA(0)31 b FF(such)f(that)300 5100 y Fz(\016)s FA(\()p FD(f)p Fz(z)t FD(g)p Fz(;)15 b(L)616 5114 y Fr(n)666 5100 y FA(\))37 b FD(\025)f Fz(\017)h(>)g FA(0)29 b FF(for)g(all)i(lar) n(ge)e Fz(n)p FF(.)46 b(Now)28 b(consider)g(the)g(ball)j Fz(B)2731 5119 y Fr(\017=)p Fs(2)2834 5100 y FA(\()p Fz(z)t FA(\))p FF(.)48 b(Let)28 b Fz(N)39 b FF(be)28 b(such)1905 5349 y(33)p eop %%Page: 34 39 34 38 bop 300 391 a FF(that)25 b(the)f(intervals)i(making)f(up)g Fz(I)1516 405 y Fr(N)1608 391 y FF(have)h(lengths)e(less)g(than)h Fz(\017=)p FA(2)p FF(.)37 b(Since)25 b Fz(z)34 b FD(2)3115 323 y Fq(T)3190 418 y Fr(n)3253 391 y Fz(I)3293 405 y Fr(n)3369 391 y FD(\023)c Fz(I)3510 405 y Fr(N)3577 391 y FF(,)300 561 y Fz(L)362 575 y Fr(N)454 561 y FD(\\)24 b Fz(B)608 579 y Fr(\017=)p Fs(2)711 561 y FA(\()p Fz(z)t FA(\))30 b FF(is)e(non-empty)-10 b(,)28 b(since)h(the)e(ball)k(must)c (cover)i(at)g(least)f(one)g(endpoint)e(of)j(the)300 730 y(particular)e(interval)g(containing)f(the)f(point)h Fz(z)t FF(.)38 b(Thus)25 b Fz(\016)s FA(\()p FD(f)p Fz(z)t FD(g)p Fz(;)15 b(L)2555 744 y Fr(N)2625 730 y FA(\))31 b Fz(<)g(\017=)p FA(2)p FF(,)d(a)e(contradiction.)300 899 y(Consequently)-10 b(,)20 b(we)i(can)i(infer)f(that)f(if)h Fz(z)30 b FD(2)1826 831 y Fq(T)1902 926 y Fr(n)1964 899 y Fz(I)2004 913 y Fr(n)2051 899 y FF(,)22 b(then)g Fz(z)29 b FD(2)p 2455 820 247 4 v 2455 831 a Fq(S)2531 926 y Fr(n)2593 899 y Fz(L)2655 913 y Fr(n)2702 899 y FF(.)583 1069 y(Thus,)p 831 990 V 831 1001 a Fq(S)907 1096 y Fr(n)969 1069 y Fz(L)1031 1083 y Fr(n)1103 1069 y FD(\022)1199 1001 y Fq(T)1275 1096 y Fr(n)1337 1069 y Fz(I)1377 1083 y Fr(n)1449 1069 y FD(\022)p 1545 990 V 1545 1001 a Fq(S)1621 1096 y Fr(n)1683 1069 y Fz(L)1745 1083 y Fr(n)1792 1069 y FF(.)e(In)c(other)e(wor)n(ds,)p 2482 990 V 2482 1001 a Fq(S)2558 1096 y Fr(n)2620 1069 y Fz(L)2682 1083 y Fr(n)2754 1069 y FA(=)2850 1001 y Fq(T)2926 1096 y Fr(n)2988 1069 y Fz(I)3028 1083 y Fr(n)3075 1069 y FF(.)p 3534 1069 4 62 v 3538 1011 55 4 v 3538 1069 V 3592 1069 4 62 v 583 1305 a(W)-5 b(ith)27 b(the)g(framework)f(set)g(up,)h(we)g(can) h(\002nally)g(pr)n(ove)e(the)g(Main)i(Theor)n(em)e(in)i(the)300 1475 y(special)23 b(case)g(of)f(a)i(left-leaning)f(transition)f(set.) 300 1680 y Fp(Theorem)h(3.17.)42 b FA(lim)1046 1694 y Fr(n)p Fn(!1)1249 1680 y FA(\000)1306 1694 y Fr(n)1378 1680 y FA(=)25 b(\003)h(=)1659 1612 y Fq(T)1734 1707 y Fr(n)1797 1680 y Fz(I)1837 1694 y Fr(n)1906 1680 y Fw(in)d(the)h(Hausdorff)g(metric.)300 1916 y(Pr)n(oof.)46 b FF(Let)19 b Fz(\017)25 b(>)g FA(0)20 b FF(be)f(arbitrary)-10 b(.)27 b(By)19 b(Lemma)g(3.15,)i(ther)n(e)e(is)g(an)h Fz(N)2594 1930 y Fs(1)2653 1916 y FF(such)f(that)g(for)g(all)i Fz(n)k(>)g(N)3538 1930 y Fs(1)3577 1916 y FF(,)1301 2131 y FA(\000)1358 2145 y Fr(n)1430 2131 y FD(\022)g Fz(B)1595 2145 y Fr(\017)1628 2131 y FA(\()p Fz(L)1725 2145 y Fr(n)1772 2131 y FA(\))1430 2325 y FD(\022)g Fz(B)1595 2339 y Fr(\017)1628 2325 y FA(\()1663 2239 y Fq([)1682 2430 y Fr(m)1779 2325 y Fz(L)1841 2339 y Fr(m)1907 2325 y FA(\))1430 2591 y FD(\022)g Fz(B)1595 2605 y Fr(\017)1628 2591 y FA(\()p 1663 2484 245 4 v -87 x Fq([)1682 2695 y Fr(m)1779 2591 y Fz(L)1841 2605 y Fr(m)1907 2591 y FA(\))1430 2856 y(=)g Fz(B)1595 2870 y Fr(\017)1628 2856 y FA(\(\003\))205 b FF(by)22 b(Lemma)h(3.16,)300 3070 y(so)f(the)g(r)n(equir)n(ement)g (\(2.31a\))j(is)e(satis\002ed.)j(As)c(well,)h(by)g(Lemma)g(3.14,)g (ther)n(e)f(is)h(an)g Fz(N)3351 3084 y Fs(0)3413 3070 y FF(such)300 3240 y(that)f(for)h(all)h Fz(n)h(>)g(N)989 3254 y Fs(0)1028 3240 y FF(,)1301 3454 y Fz(B)1370 3468 y Fr(\017)1403 3454 y FA(\(\000)1495 3468 y Fr(n)1542 3454 y FA(\))g FD(\023)g Fz(I)1738 3468 y Fr(n)1602 3649 y FD(\023)1698 3562 y Fq(\\)1718 3753 y Fr(m)1814 3649 y Fz(I)1854 3663 y Fr(m)1602 3914 y FA(=)g(\003)205 b FF(by)22 b(Lemma)h(3.16,)300 4128 y(so)i(the)g(r)n(equir)n(ement)f (\(2.31b\))k(is)d(also)h(satis\002ed.)35 b(This)26 b(implies)g(that)f FA(\000)2811 4142 y Fr(n)2888 4128 y FD(!)31 b FA(\003)26 b FF(in)g(the)e(Haus-)300 4298 y(dorf)n(f)f(metric,)f(as)h Fz(n)i FD(!)g(1)p FF(.)p 3534 4298 4 62 v 3538 4240 55 4 v 3538 4298 V 3592 4298 4 62 v 300 4663 a Ft(3.4)119 b(Right-leaning)29 b(T)-11 b(ransition)29 b(Sets)300 4931 y FF(Compar)n(ed)i(to)f(the)g(pr)n(evious)g(case)h(of)g (left-leaning)h(transition)e(sets,)h(this)g(second)f(special)300 5100 y(case)h(is)g(easier)f(to)h(deal)g(with.)52 b(Intuitively)-10 b(,)32 b(starting)e(a)i(random)e(set-walk)h(near)f(the)h(right)1905 5349 y(34)p eop %%Page: 35 40 35 39 bop 300 391 a FF(absorbing)21 b(barrier)-7 b(,)21 b(and)g(then)f(moving)h(about)f(as)h(dictated)f(by)h(a)g(set)e(of)i (right-leaning)g(tran-)300 561 y(sition)e(triples)g(should)f(r)n(esult) g(in)i(a)g(high)f(pr)n(obability)i(in)e(being)h(ultimately)f(absorbed)g (by)g(the)300 730 y(right)27 b(barrier)-7 b(.)42 b(In)26 b(fact,)j(as)e Fz(n)g FF(tends)e(to)i(in\002nity)-10 b(,)28 b(all)g(such)f(pr)n(obabilities)h(should)f(tend)e(to)i FA(1)p FF(.)300 899 y(W)-8 b(e)22 b(show)f(the)h(corr)n(ectness)f(of)i (this)f(r)n(easoning)g(in)h(this)g(section.)583 1069 y(Let)f FA(1)p Fz(=)p FA(2)28 b FD(\024)e Fz(y)1040 1083 y Fs(1)1105 1069 y Fz(<)f(y)1246 1083 y Fs(2)1311 1069 y Fz(<)h FD(\001)15 b(\001)g(\001)27 b Fz(<)f(y)1682 1084 y Fr(d)1748 1069 y Fz(<)g FA(1)e FF(be)f(a)g(non-empty)-10 b(,)21 b(\002nite)i(set)f(of)h(r)n(eal)h(numbers,)300 1238 y(and)1383 1408 y FD(T)48 b FA(=)25 b FD(f)p FB(T)1684 1422 y Fr(i)1738 1408 y FA(=)g FB(T)p FA(\()p Fz(y)1976 1422 y Fr(i)2004 1408 y FA(\))e FD(j)g Fz(i)i FD(2)g FA([1)h(:)g Fz(d)p FA(])p FD(g)300 1608 y FF(be)d(a)g(transition)f (set.)583 1777 y(For)g(each)h Fz(y)992 1791 y Fr(i)1020 1777 y FF(,)f(de\002ne)g(the)g(uniform)h(strategy)d(sequence)h FA(\()p FB(T)2675 1733 y Fr(j)2675 1804 y(i)2712 1777 y FA(\))2747 1791 y Fr(j)t Fn(2)p Fl(N)2904 1777 y FA(=)k Fz(U)10 b FA(\()p FB(T)3169 1791 y Fr(i)3197 1777 y FA(\))p FF(.)300 1983 y Fp(Theorem)23 b(3.18.)42 b FA(lim)1046 1997 y Fr(n)p Fn(!1)1249 1983 y FA(\000)1306 1997 y Fr(n)1378 1983 y FA(=)25 b FD(f)p FA(1)p FD(g)p Fw(.)300 2220 y(Pr)n(oof.)46 b FF(For)35 b(each)g(strategy)e Fz(s)48 b FD(2)f Fm(S)1567 2238 y Fs([)o(1:)p Fr(n)p Fs(])1702 2220 y FA(\()p FD(T)23 b FA(\))p FF(,)38 b Fz(c)p FA([)p FB(T)2032 2187 y Fr(n)2032 2244 y Fs(1)2080 2220 y FA(])48 b FD(\024)g Fz(c)p FA([)p Fz(s)p FA(])g FD(\024)g Fz(c)p FA([)p FB(T)2697 2187 y Fr(n)2697 2247 y(d)2745 2220 y FA(])35 b FF(by)g(the)f(Comparison)300 2389 y(Lemma.)66 b(Mor)n(eover)-7 b(,)37 b(for)e(all)i Fz(i)49 b FD(2)f FA([1)i(:)f Fz(d)p FA(])p FF(,)38 b Fz(c)p FA([)p FB(T)2069 2356 y Fr(n)2069 2415 y(i)2117 2389 y FA(])49 b FD(!)f FA(1)36 b FF(as)f Fz(n)49 b FD(!)f(1)35 b FF(by)h(the)e(gambler)7 b('s)300 2558 y(r)o(uin)30 b(r)n(esult)f(\(2.12\))j(\(for)e(both)f(the)g(case)h Fz(y)1793 2572 y Fs(1)1871 2558 y FA(=)38 b(1)p Fz(=)p FA(2)31 b FF(and)f(the)f(case)h Fz(y)2729 2572 y Fs(1)2807 2558 y Fz(>)38 b FA(1)p Fz(=)p FA(2)p FF(\).)51 b(Thus)29 b FA(\000)3444 2572 y Fr(n)3529 2558 y FD(\022)300 2728 y FA([)p Fz(c)p FA([)p FB(T)451 2695 y Fr(n)451 2752 y Fs(1)499 2728 y FA(])p Fz(;)15 b(c)p FA([)p FB(T)690 2695 y Fr(n)690 2755 y(d)738 2728 y FA(]])23 b FF(implies)g(that)f FA(lim)1437 2742 y Fr(n)p Fn(!1)1640 2728 y FA(\000)1697 2742 y Fr(n)1769 2728 y FD(\022)j(f)p FA(1)p FD(g)p FF(.)k(As)22 b FA(lim)14 b(\000)2383 2742 y Fr(n)2455 2728 y FD(6)p FA(=)25 b Fz(\036)p FF(,)d FA(lim)14 b(\000)2849 2742 y Fr(n)2921 2728 y FA(=)25 b FD(f)p FA(1)p FD(g)p FF(.)p 3534 2728 4 62 v 3538 2670 55 4 v 3538 2728 V 3592 2728 4 62 v 300 3093 a Ft(3.5)119 b(Mixed)30 b(T)-11 b(ransition)29 b(Sets)300 3361 y FF(In)d(this)f(thir)n(d)h(special)g(case)g(of)g(the)f (Main)i(Theor)n(em,)f(we)f(appr)n(oach)i(the)e(pr)n(oblem)h(of)g(pr)n (ov-)300 3530 y(ing)j(the)g(conver)n(gence)f(r)n(esult)h(fr)n(om)h(a)g (dif)n(fer)n(ent)f(angle.)48 b(W)-8 b(e)28 b(will)j(actually)g(specify) d(at)i(the)300 3700 y(outset)25 b(the)g(candidate)i(for)g(the)f(limit)i (set)d FA(\003)p FF(,)j(and)e(then)g(pr)n(oceed)f(to)i(pr)n(ove)f(that) g(the)g(candi-)300 3869 y(date)j(is)h(indeed)e(the)h(corr)n(ect)h (limit)h(set.)48 b(The)29 b(method)f(used)h(to)g(pr)n(ove)g(the)g(corr) n(ectness)f(of)300 4038 y(the)h(candidate)g(mimics)i(that)f(of)f(a)i (gr)n(eedy)c(algorithm,)32 b(wher)n(e)d(we)g(show)f(that)i(given)f(any) 300 4208 y(point)20 b Fz(\025)h FF(in)g(the)e(candidate)i(set)e FA(\003)p FF(,)i(we)f(can)h(systematically)g(constr)o(uct)f(a)h (sequence)e(of)h(points)300 4377 y FA(\()p Fz(c)p FA([)p Fz(s)442 4344 y Fr(\025)442 4399 y(n)490 4377 y FA(]\))550 4391 y Fr(n)p Fn(2)p Fl(N)692 4377 y FF(,)34 b(wher)n(e)d Fz(c)p FA([)p Fz(s)1138 4344 y Fr(\025)1138 4399 y(n)1185 4377 y FA(])42 b FD(2)g FA(\000)1412 4391 y Fr(n)1459 4377 y FF(,)33 b(conver)n(ging)f(to)f Fz(\025)p FF(.)55 b(Hence)31 b(we)g(may)h(conclude)f(that)h(the)300 4546 y(candidate)23 b(limit)h(is)e(\223close\224)i(to)e(the)f(sets)g FA(\000)1810 4560 y Fr(n)1880 4546 y FF(for)h(suf)n(\002ciently)h(lar)n (ge)g Fz(n)p FF(.)583 4716 y(Let)30 b FA(0)40 b Fz(<)f(x)990 4730 y Fs(1)1069 4716 y Fz(<)g(x)1231 4730 y Fs(2)1310 4716 y Fz(<)g FD(\001)15 b(\001)g(\001)40 b Fz(<)g(x)1728 4731 y Fr(d)1807 4716 y Fz(<)g FA(1)p FF(,)32 b(wher)n(e)e Fz(x)2351 4730 y Fs(1)2429 4716 y Fz(<)40 b FA(1)p Fz(=)p FA(2)h Fz(<)e(x)2878 4731 y Fr(d)2949 4716 y FF(and)30 b Fz(d)40 b(>)f FA(1)p FF(,)32 b(be)f(a)300 4885 y(non-empty)-10 b(,)21 b(\002nite)h(set)f(of)i(r)n(eal)h(numbers,)d(and)1380 5100 y FD(T)47 b FA(=)25 b FD(f)p FB(T)1680 5114 y Fr(i)1734 5100 y FA(=)g FB(T)p FA(\()p Fz(x)1979 5114 y Fr(i)2007 5100 y FA(\))f FD(j)f Fz(i)i FD(2)g FA([1)h(:)f Fz(d)p FA(])p FD(g)1905 5349 y FF(35)p eop %%Page: 36 41 36 40 bop 300 391 a FF(be)23 b(a)g(transition)f(set.)583 561 y(For)g(each)h Fz(x)999 575 y Fr(i)1027 561 y FF(,)g(de\002ne)e (the)h(uniform)h(strategy)d(sequence)h FA(\()p FB(T)2682 516 y Fr(j)2682 588 y(i)2719 561 y FA(\))2754 575 y Fr(j)t Fn(2)p Fl(N)2911 561 y FA(=)k Fz(U)10 b FA(\()p FB(T)p FA(\()p Fz(x)3263 575 y Fr(i)3292 561 y FA(\)\))p FF(.)300 805 y Fp(Proposition)22 b(3.19.)42 b Fw(The)23 b(candidate)i(for)e(the) h(limit)g(set)g FA(\003)e Fw(is)i(the)g(closed)g(interval)1522 1065 y FA(\003)h(=)1706 965 y Fq(h)1788 1065 y FA(lim)1749 1120 y Fr(m)p Fn(!1)1968 1065 y Fz(c)p FA([)p FB(T)2094 1028 y Fr(m)2094 1088 y Fs(1)2161 1065 y FA(])p Fz(;)15 b FA(1)2271 965 y Fq(i)2353 1065 y Fz(:)1002 b FF(\(3.10\))583 1326 y(W)-5 b(ith)32 b(the)e(existing)g(r)n(esults)g(pr)n(oven)h(in)h (Section)e(3.2,)k(we)d(can)h(alr)n(eady)f(satisfy)g(the)300 1496 y(condition)22 b(\(2.31a\))r(,)h(which)g(is)g(half)h(of)e(the)g (conver)n(gence)g(ar)n(gument.)300 1740 y Fp(Lemma)h(3.20.)42 b FA(lim)980 1754 y Fr(n)p Fn(!1)1184 1740 y Fz(\016)s FA(\(\000)1319 1754 y Fr(n)1367 1740 y Fz(;)15 b FA(\003\))26 b(=)f(0)p Fw(.)300 1984 y(Pr)n(oof.)46 b FF(By)26 b(the)g(Comparison)h (Lemma,)h Fz(c)p FA([)p Fz(s)p FA(])33 b FD(\025)g Fz(c)p FA([)p FB(T)2125 1951 y Fr(n)2125 2008 y Fs(1)2172 1984 y FA(])28 b FF(for)e(all)j Fz(s)j FD(2)h Fm(S)2718 2002 y Fs([1:)p Fr(n)o Fs(])2853 1984 y FA(\()p FD(T)23 b FA(\))p FF(,)28 b(implying)f(that)300 2153 y FA(\000)357 2167 y Fr(n)429 2153 y FD(\022)e FA([)p Fz(c)p FA([)p FB(T)676 2120 y Fr(n)676 2177 y Fs(1)724 2153 y FA(])p Fz(;)15 b FA(1])q FF(.)28 b(By)21 b(Pr)n(oposition)g(3.4,)i Fz(c)p FA([)p FB(T)1813 2120 y Fr(n)1813 2177 y Fs(1)1860 2153 y FA(])g FF(conver)n(ges)e(to)g FA(lim)2563 2167 y Fr(m)p Fn(!1)2786 2153 y Fz(c)p FA([)p FB(T)2912 2120 y Fr(m)2912 2177 y Fs(1)2979 2153 y FA(])k(=)g Fz(x)3177 2167 y Fs(1)3216 2153 y Fz(=)p FA(\(1)20 b FD(\000)f Fz(x)3503 2167 y Fs(1)3542 2153 y FA(\))p FF(.)300 2322 y(Hence)i Fz(\016)s FA(\(\000)713 2336 y Fr(n)761 2322 y Fz(;)15 b FA(\003\))26 b FD(\024)f Fz(\016)s FA(\([)q Fz(c)p FA([)p FB(T)1251 2289 y Fr(n)1251 2347 y Fs(1)1299 2322 y FA(])p Fz(;)15 b FA(1])h Fz(;)f FA(\003\))27 b(=)d(\()q(lim)1872 2336 y Fr(m)p Fn(!1)2095 2322 y Fz(c)p FA([)p FB(T)2221 2289 y Fr(m)2221 2347 y Fs(1)2288 2322 y FA(]\))18 b FD(\000)f Fz(c)p FA([)p FB(T)2580 2289 y Fr(n)2580 2347 y Fs(1)2627 2322 y FA(])p FF(,)22 b(which)g(appr)n(oaches)g FA(0)g FF(as)300 2492 y Fz(n)j FD(!)g(1)p FF(.)j(In)22 b(other)f(wor)n(ds,)h FA(lim)1404 2506 y Fr(n)p Fn(!1)1607 2492 y Fz(\016)s FA(\(\000)1742 2506 y Fr(n)1790 2492 y Fz(;)15 b FA(\003\))26 b(=)f(0)p FF(.)p 3534 2492 4 62 v 3538 2434 55 4 v 3538 2492 V 3592 2492 4 62 v 583 2736 a(Let)d(us)g(de\002ne)f(the)h(function)1340 2997 y Fz(cl)r FA(\()p Fz(s)p FA(\))k(=)64 b(lim)1643 3051 y Fr(m)p Fn(!1)1862 2997 y Fz(c)p FA([)p FB(T)1988 2959 y Fr(m)1988 3019 y Fs(1)2055 2997 y Fz(s)p FA(])1547 3292 y(=)25 b Fz(c)p FA([)p Fz(s)p FA(])c(+)1897 3198 y Fz(a)p FA([)p Fz(s)p FA(])2053 3097 y Fq(\020)2162 3161 y Fr(x)2202 3170 y Fk(1)p 2117 3177 165 4 v 2117 3229 a Fs(1)p Fn(\000)p Fr(x)2247 3238 y Fk(1)2292 3097 y Fq(\021)2361 3198 y Fz(d)p FA([)p Fz(s)p FA(])p 1897 3272 605 4 v 1901 3387 a(1)f FD(\000)g Fz(b)p FA([)p Fz(s)p FA(])2204 3286 y Fq(\020)2314 3350 y Fr(x)2354 3359 y Fk(1)p 2269 3366 165 4 v 2269 3418 a Fs(1)p Fn(\000)p Fr(x)2399 3427 y Fk(1)2443 3286 y Fq(\021)2534 3292 y Fz(:)826 b FF(\(3.1)-5 b(1\))300 3616 y(Intuitively)-10 b(,)22 b Fz(cl)r FA(\()p Fz(s)p FA(\))h FF(r)n(epr)n(esents)d(a)j (lower)f(bound)g(for)h Fz(c)p FA([)p Fz(s)2252 3583 y Fn(0)2275 3616 y Fz(s)p FA(])g FF(for)f(any)h(\223in\002nite\224)g (strategy)e Fz(s)3466 3583 y Fn(0)3489 3616 y FF(.)28 b(It)300 3785 y(will)c(be)e(used)g(extensively)f(in)i(the)f(gr)n(eedy)e (algorithm.)583 3954 y(Below)-8 b(,)34 b(we)e(pr)n(ove)g(thr)n(ee)g(r)n (esults)f(about)i(rates)e(of)i(conver)n(gence)f(r)n(elated)g(to)g(char) n(-)300 4124 y(acteristic)j(pr)n(obabilities,)j(to)c(be)g(used)f(later) h(in)h(pr)n(oving)f(the)g(corr)n(ectness)e(of)j(the)e(gr)n(eedy)300 4293 y(algorithm.)300 4537 y Fp(Proposition)22 b(3.21.)42 b Fw(For)23 b(any)g(strategy)j Fz(s)e FD(2)h Fm(S)o FA(\()p FD(T)17 b FA(\))23 b Fw(wher)n(e)h Fz(c)p FA([)p Fz(s)p FA(])i Fz(<)f FA(1)p Fw(,)1446 4798 y FA(lim)1407 4852 y Fr(m)p Fn(!1)1626 4798 y Fz(c)p FA([)p Fz(s)p FB(T)1795 4760 y Fr(m)1795 4820 y Fs(1)1861 4798 y FA(])h(=)64 b(lim)2008 4852 y Fr(m)p Fn(!1)2227 4798 y Fz(c)p FA([)p FB(T)2353 4760 y Fr(m)2353 4820 y Fs(1)2420 4798 y FA(])23 b Fz(:)887 b FF(\(3.12\))1905 5349 y(36)p eop %%Page: 37 42 37 41 bop 300 391 a Fw(Pr)n(oof.)46 b FF(Let)300 652 y Fz(')359 667 y Fd(T)407 644 y Fj(m)407 688 y Fk(1)496 652 y FA(=)25 b FD(h)p Fz(a;)15 b(b;)g(c;)g(d)p FD(i)496 895 y FA(=)592 740 y Fq(*)675 819 y FA(\()720 782 y Fs(1)p Fn(\000)p Fr(x)850 791 y Fk(1)p 720 798 165 4 v 765 850 a Fr(x)805 859 y Fk(1)895 819 y FA(\))930 786 y Fr(m)p Fs(+1)1107 819 y FD(\000)20 b FA(\()1243 782 y Fs(1)p Fn(\000)p Fr(x)1373 791 y Fk(1)p 1243 798 V 1288 850 a Fr(x)1328 859 y Fk(1)1418 819 y FA(\))1453 786 y Fr(m)p 675 874 845 4 v 813 965 a FA(\()858 928 y Fs(1)p Fn(\000)p Fr(x)988 937 y Fk(1)p 858 944 165 4 v 903 997 a Fr(x)943 1006 y Fk(1)1033 965 y FA(\))1068 939 y Fr(m)p Fs(+1)1245 965 y FD(\000)g FA(1)1529 895 y Fz(;)1580 819 y FA(\()1625 782 y Fs(1)p Fn(\000)p Fr(x)1755 791 y Fk(1)p 1625 798 V 1670 850 a Fr(x)1710 859 y Fk(1)1800 819 y FA(\))1835 786 y Fr(m)p Fs(+1)2012 819 y FD(\000)2113 782 y Fs(1)p Fn(\000)p Fr(x)2243 791 y Fk(1)p 2113 798 V 2158 850 a Fr(x)2198 859 y Fk(1)p 1580 874 708 4 v 1649 965 a FA(\()1694 928 y Fs(1)p Fn(\000)p Fr(x)1824 937 y Fk(1)p 1695 944 165 4 v 1740 997 a Fr(x)1780 1006 y Fk(1)1869 965 y FA(\))1904 939 y Fr(m)p Fs(+1)2081 965 y FD(\000)g FA(1)2297 895 y Fz(;)2348 819 y FA(\()2393 782 y Fs(1)p Fn(\000)p Fr(x)2523 791 y Fk(1)p 2393 798 V 2438 850 a Fr(x)2478 859 y Fk(1)2567 819 y FA(\))2602 786 y Fr(m)p Fs(+1)2780 819 y FD(\000)g FA(1)p 2348 874 569 4 v 2348 965 a(\()2393 928 y Fs(1)p Fn(\000)p Fr(x)2523 937 y Fk(1)p 2393 944 165 4 v 2438 997 a Fr(x)2478 1006 y Fk(1)2567 965 y FA(\))2602 939 y Fr(m)p Fs(+1)2780 965 y FD(\000)g FA(1)2926 895 y Fz(;)3100 782 y Fs(1)p Fn(\000)p Fr(x)3230 791 y Fk(1)p 3100 798 V 3145 850 a Fr(x)3185 859 y Fk(1)3295 819 y FD(\000)g FA(1)p 2976 874 569 4 v 2976 965 a(\()3021 928 y Fs(1)p Fn(\000)p Fr(x)3151 937 y Fk(1)p 3022 944 165 4 v 3067 997 a Fr(x)3107 1006 y Fk(1)3196 965 y FA(\))3231 939 y Fr(m)p Fs(+1)3408 965 y FD(\000)g FA(1)3555 740 y Fq(+)3666 895 y Fz(:)300 1175 y FF(Thus)1153 1435 y Fz(c)p FA([)p Fz(s)p FB(T)1322 1398 y Fr(m)1322 1458 y Fs(1)1388 1435 y FA(])26 b(=)f Fz(c)20 b FA(+)1723 1374 y Fz(a)i(c)p FA([)p Fz(s)p FA(])h Fz(d)p 1695 1414 329 4 v 1695 1498 a FA(1)e FD(\000)f Fz(c)p FA([)p Fz(s)p FA(])p Fz(b)1439 1678 y FA(=)25 b Fz(c)20 b FA(+)g Fz(a)j(c)p FA([)p Fz(s)p FA(])2157 1617 y Fz(d)p 1898 1657 564 4 v 1898 1741 a FA(1)e FD(\000)f Fz(c)p FA([)p Fz(s)p FA(]\(1)h FD(\000)f Fz(d)p FA(\))1439 1945 y(=)25 b Fz(c)20 b FA(+)g Fz(a)j(c)p FA([)p Fz(s)p FA(])1903 1817 y Fq(\022)1980 1883 y FA(1)e FD(\000)f Fz(c)p FA([)p Fz(s)p FA(])p 1980 1924 290 4 v 2101 2007 a Fz(d)2300 1945 y FA(+)g Fz(c)p FA([)p Fz(s)p FA(])2523 1817 y Fq(\023)2590 1839 y Fn(\000)p Fs(1)2722 1945 y Fz(:)300 2252 y FF(As)27 b(well,)j(we)e(have)1031 2151 y Fq(\020)1095 2207 y Fs(1)p Fn(\000)p Fr(c)p Fs([)p Fr(s)p Fs(])p 1095 2231 193 4 v 1174 2283 a Fr(d)1318 2252 y FA(+)20 b Fz(c)p FA([)p Fz(s)p FA(])1541 2151 y Fq(\021)1596 2174 y Fn(\000)p Fs(1)1726 2252 y FD(!)35 b FA(0)29 b FF(as)f Fz(m)35 b FD(!)g(1)p FF(,)30 b(since)e Fz(c)p FA([)p Fz(s)p FA(])36 b Fz(<)f FA(1)p FF(,)29 b Fz(x)3076 2266 y Fs(1)3151 2252 y Fz(<)35 b FA(1)p Fz(=)p FA(2)p FF(,)c(and)300 2421 y Fz(d)395 2370 y(:)373 2421 y FA(=)25 b Fz(d)p FA([)p FB(T)603 2388 y Fr(m)603 2446 y Fs(1)670 2421 y FA(])g FD(!)g FA(0)e FF(as)g Fz(m)i FD(!)g(1)p FF(.)j(Hence)1051 2682 y FA(lim)1012 2737 y Fr(m)p Fn(!1)1231 2682 y FD(j)p Fz(c)p FA([)p Fz(s)p FB(T)1425 2644 y Fr(m)1425 2704 y Fs(1)1492 2682 y FA(])20 b FD(\000)g Fz(c)p FD(j)26 b FA(=)63 b(lim)1814 2737 y Fr(m)p Fn(!1)2033 2682 y FD(j)p Fz(c)p FA([)p Fz(s)p FB(T)2227 2644 y Fr(m)2227 2704 y Fs(1)2293 2682 y FA(])21 b FD(\000)f Fz(c)p FA([)p FB(T)2556 2644 y Fr(m)2556 2704 y Fs(1)2623 2682 y FA(])p FD(j)26 b FA(=)f(0)e Fz(;)300 2943 y FF(yielding)1446 3112 y FA(lim)1407 3167 y Fr(m)p Fn(!1)1626 3112 y Fz(c)p FA([)p Fz(s)p FB(T)1795 3074 y Fr(m)1795 3134 y Fs(1)1861 3112 y FA(])j(=)64 b(lim)2008 3167 y Fr(m)p Fn(!1)2227 3112 y Fz(c)p FA([)p FB(T)2353 3074 y Fr(m)2353 3134 y Fs(1)2420 3112 y FA(])23 b Fz(:)p 3534 3335 4 62 v 3538 3277 55 4 v 3538 3335 V 3592 3335 4 62 v 300 3579 a Fp(Proposition)f(3.22.)42 b Fw(For)23 b(any)g(strategy)j Fz(s)e FD(2)h Fm(S)o FA(\()p FD(T)17 b FA(\))p Fw(,)1500 3840 y FA(lim)1461 3895 y Fr(m)p Fn(!1)1680 3840 y Fz(cl)r FA(\()p FB(T)1845 3854 y Fr(j)1882 3840 y FB(T)1944 3803 y Fr(m)1944 3863 y Fs(1)2010 3840 y Fz(s)p FA(\))25 b(=)g Fz(cl)r FA(\()p Fz(s)p FA(\))f Fz(:)941 b FF(\(3.13\))300 4101 y Fw(Pr)n(oof.)46 b FF(First,)22 b(let)g Fz(s)955 4068 y Fn(0)955 4123 y Fr(n)1027 4101 y FA(=)j FB(T)1185 4068 y Fr(n)1185 4125 y Fs(1)1232 4101 y FB(T)1294 4115 y Fr(j)1330 4101 y FF(,)e(and)f(notice)h(that)679 4361 y Fz(cl)r FA(\()p FB(T)844 4375 y Fr(j)881 4361 y FB(T)943 4324 y Fr(m)943 4384 y Fs(1)1009 4361 y FA(\))j(=)53 b(lim)1166 4416 y Fr(n)p Fn(!1)1365 4361 y Fz(c)p FA([)p FB(T)1491 4324 y Fr(n)1491 4384 y Fs(1)1538 4361 y FB(T)1600 4375 y Fr(j)1637 4361 y FB(T)1699 4324 y Fr(m)1699 4384 y Fs(1)1765 4361 y FA(])1070 4590 y(=)g(lim)1166 4644 y Fr(n)p Fn(!1)1365 4590 y Fz(c)p FA([)p FB(T)1491 4552 y Fr(m)1491 4612 y Fs(1)1558 4590 y FA(])20 b(+)1704 4528 y Fz(a)p FA([)p FB(T)1839 4495 y Fr(m)1839 4553 y Fs(1)1906 4528 y FA(])p Fz(c)p FA([)p Fz(s)2038 4495 y Fn(0)2038 4551 y Fr(n)2085 4528 y FA(])p Fz(d)p FA([)p FB(T)2244 4495 y Fr(m)2244 4553 y Fs(1)2312 4528 y FA(])p 1704 4569 633 4 v 1743 4652 a(1)h FD(\000)f Fz(c)p FA([)p Fz(s)2007 4626 y Fn(0)2007 4675 y Fr(n)2054 4652 y FA(])p Fz(b)p FA([)p FB(T)2205 4621 y Fr(m)2205 4678 y Fs(1)2272 4652 y FA(])1070 4841 y(=)25 b Fz(c)p FA([)p FB(T)1292 4803 y Fr(m)1292 4863 y Fs(1)1359 4841 y FA(])20 b(+)1505 4780 y Fz(a)p FA([)p FB(T)1640 4747 y Fr(m)1640 4804 y Fs(1)1707 4780 y FA(])p Fz(c)1771 4794 y Fn(1)1846 4780 y Fz(d)p FA([)p FB(T)1980 4747 y Fr(m)1980 4804 y Fs(1)2047 4780 y FA(])p 1505 4820 568 4 v 1544 4903 a(1)h FD(\000)f Fz(c)1740 4917 y Fn(1)1815 4903 y Fz(b)p FA([)p FB(T)1941 4872 y Fr(m)1941 4929 y Fs(1)2008 4903 y FA(])2287 4841 y FF(wher)n(e)i Fz(c)2599 4855 y Fn(1)2699 4841 y FA(=)54 b(lim)2795 4896 y Fr(n)p Fn(!1)2994 4841 y Fz(c)p FA([)p Fz(s)3101 4803 y Fn(0)3101 4863 y Fr(n)3148 4841 y FA(])23 b Fz(:)1905 5349 y FF(37)p eop %%Page: 38 43 38 42 bop 300 391 a FF(Hence,)29 b(since)f Fz(c)878 405 y Fn(1)990 391 y Fz(<)36 b FA(1)29 b FF(by)g(the)e(natur)n(e)i(of)f (the)g(sequence)f Fz(s)2448 358 y Fn(0)2448 414 y Fr(n)2531 391 y FA(=)36 b FB(T)2700 358 y Fr(n)2700 416 y Fs(1)2747 391 y FB(T)2809 405 y Fr(j)2845 391 y FF(,)30 b(we)e(can)h(apply)g(the) 300 561 y(r)n(esult)22 b(derived)g(in)h(the)f(pr)n(oof)g(of)g(the)g(pr) n(evious)g(pr)n(oposition)f(and)i(get)1393 821 y FA(lim)1355 876 y Fr(m)p Fn(!1)1573 821 y Fz(cl)r FA(\()p FB(T)1738 835 y Fr(j)1775 821 y FB(T)1837 784 y Fr(m)1837 844 y Fs(1)1904 821 y FA(\))i(=)64 b(lim)2060 876 y Fr(m)p Fn(!1)2279 821 y Fz(c)p FA([)p FB(T)2405 784 y Fr(m)2405 844 y Fs(1)2472 821 y FA(])23 b Fz(:)300 1082 y FF(By)f(Pr)n(oposition) f(3.5,)i(we)f(can)i(conclude)1199 1343 y FA(lim)1161 1397 y Fr(m)p Fn(!1)1379 1343 y Fz(cl)r FA(\()p FB(T)1544 1357 y Fr(j)1581 1343 y FB(T)1643 1305 y Fr(m)1643 1365 y Fs(1)1710 1343 y Fz(s)p FA(\))h(=)64 b(lim)1909 1397 y Fr(m)p Fn(!1)2128 1343 y Fz(c)p FA([)p FB(T)2254 1305 y Fr(m)2254 1365 y Fs(1)2321 1343 y Fz(s)p FA(])25 b(=)g Fz(cl)r FA(\()p Fz(s)p FA(\))e Fz(:)p 3534 1603 4 62 v 3538 1545 55 4 v 3538 1603 V 3592 1603 4 62 v 300 1848 a Fp(Proposition)f(3.23.)42 b Fw(Ther)n(e)24 b(is)f(a)g Fz(K)32 b FD(2)25 b FA(\(0)p Fz(;)15 b FA(1])25 b Fw(such)e(that,)h (for)g(any)f(strategy)i Fz(s)g FD(2)g Fm(S)o FA(\()p FD(T)17 b FA(\))p Fw(,)1401 2047 y Fz(cl)r FA(\()p FB(T)1566 2062 y Fr(d)1607 2047 y FB(T)1669 2061 y Fs(1)1709 2047 y Fz(s)p FA(\))j FD(\000)g Fz(cl)r FA(\()p Fz(s)p FA(\))p 1401 2087 678 4 v 1452 2171 a Fz(cl)r FA(\()p FB(T)1617 2186 y Fr(d)1658 2171 y Fz(s)p FA(\))g FD(\000)g Fz(cl)r FA(\()p Fz(s)p FA(\))2114 2108 y FD(\025)25 b Fz(K)32 b(>)25 b FA(0)e Fz(:)872 b FF(\(3.14\))300 2369 y Fw(Pr)n(oof.)46 b FF(Let)22 b Fz(')774 2383 y Fr(s)836 2369 y FA(=)j FD(h)p Fz(a;)15 b(b;)g(c;)g(d)p FD(i)p FF(,)25 b(then)704 2630 y Fz(cl)r FA(\()p FB(T)869 2645 y Fr(d)910 2630 y FB(T)972 2644 y Fs(1)1011 2630 y Fz(s)p FA(\))20 b FD(\000)g Fz(cl)r FA(\()p Fz(s)p FA(\))26 b(=)1503 2501 y Fq(\022)1570 2630 y Fz(c)20 b FA(+)1757 2568 y Fz(a)j(cl)r FA(\()p FB(T)1993 2583 y Fr(d)2034 2568 y FB(T)2096 2582 y Fs(1)2136 2568 y FA(\))g Fz(d)p 1730 2609 539 4 v 1730 2692 a FA(1)e FD(\000)f Fz(cl)r FA(\()p FB(T)2052 2707 y Fr(d)2093 2692 y FB(T)2155 2706 y Fs(1)2194 2692 y FA(\))p Fz(b)2278 2501 y Fq(\023)2365 2630 y FD(\000)2456 2501 y Fq(\022)2523 2630 y Fz(c)h FA(+)2711 2568 y Fz(a)i(cl)r FA(\()p FB(T)2947 2582 y Fs(1)2987 2568 y FA(\))g Fz(d)p 2684 2609 436 4 v 2684 2692 a FA(1)d FD(\000)g Fz(cl)r FA(\()p FB(T)3005 2706 y Fs(1)3045 2692 y FA(\))p Fz(b)3129 2501 y Fq(\023)1407 2879 y FA(=)1540 2817 y Fz(a)j(cl)r FA(\()p FB(T)1776 2832 y Fr(d)1817 2817 y FB(T)1879 2831 y Fs(1)1918 2817 y FA(\))g Fz(d)p 1513 2858 539 4 v 1513 2941 a FA(1)d FD(\000)g Fz(cl)r FA(\()p FB(T)1834 2956 y Fr(d)1875 2941 y FB(T)1937 2955 y Fs(1)1977 2941 y FA(\))p Fz(b)2081 2879 y FD(\000)2209 2817 y Fz(a)j(cl)r FA(\()p FB(T)2445 2831 y Fs(1)2485 2817 y FA(\))g Fz(d)p 2182 2858 436 4 v 2182 2941 a FA(1)e FD(\000)f Fz(cl)r FA(\()p FB(T)2504 2955 y Fs(1)2543 2941 y FA(\))p Fz(b)1407 3127 y FD(\025)1513 3065 y Fz(ad)p FA(\()p Fz(cl)r FA(\()p FB(T)1808 3080 y Fr(d)1849 3065 y FB(T)1911 3079 y Fs(1)1951 3065 y FA(\))g FD(\000)g Fz(cl)r FA(\()p FB(T)h FD(\000)f FA(1\)\))p 1513 3106 978 4 v 1732 3189 a(1)h FD(\000)f Fz(cl)r FA(\()p FB(T)2054 3204 y Fr(d)2095 3189 y FB(T)2157 3203 y Fs(1)2196 3189 y FA(\))p Fz(b)300 3412 y FF(\(because)j Fz(cl)r FA(\()p FB(T)835 3427 y Fr(d)875 3412 y FB(T)937 3426 y Fs(1)977 3412 y FA(\))i FD(\025)g Fz(cl)r FA(\()p FB(T)1298 3426 y Fs(1)1338 3412 y FA(\))e FF(by)g(the)f(Comparison)g (Lemma\))1407 3698 y FD(\025)j Fz(ad)p FA(\()p Fz(cl)r FA(\()p FB(T)1798 3713 y Fr(d)1839 3698 y FB(T)1901 3712 y Fs(1)1941 3698 y FA(\))20 b FD(\000)g Fz(cl)r FA(\()p FB(T)h FD(\000)f FA(1\)\))k Fz(;)300 3984 y FF(and)805 4269 y Fz(cl)r FA(\()p FB(T)970 4284 y Fr(d)1011 4269 y Fz(s)p FA(\))c FD(\000)g Fz(cl)r FA(\()p Fz(s)p FA(\))26 b(=)1503 4141 y Fq(\022)1570 4269 y Fz(c)20 b FA(+)1757 4208 y Fz(a)j(cl)r FA(\()p FB(T)1993 4223 y Fr(d)2034 4208 y FA(\))g Fz(d)p 1730 4248 437 4 v 1730 4332 a FA(1)e FD(\000)f Fz(cl)r FA(\()p FB(T)2052 4347 y Fr(d)2093 4332 y FA(\))p Fz(b)2177 4141 y Fq(\023)2264 4269 y FD(\000)2355 4141 y Fq(\022)2422 4269 y Fz(c)g FA(+)2610 4208 y Fz(a)i(cl)r FA(\()p FB(T)2845 4222 y Fs(1)2885 4208 y FA(\))h Fz(d)p 2582 4248 436 4 v 2582 4332 a FA(1)e FD(\000)f Fz(cl)r FA(\()p FB(T)2904 4346 y Fs(1)2944 4332 y FA(\))p Fz(b)3028 4141 y Fq(\023)1407 4518 y FA(=)1540 4457 y Fz(a)j(cl)r FA(\()p FB(T)1776 4472 y Fr(d)1817 4457 y FA(\))g Fz(d)p 1513 4498 437 4 v 1513 4581 a FA(1)d FD(\000)g Fz(cl)r FA(\()p FB(T)1834 4596 y Fr(d)1875 4581 y FA(\))p Fz(b)1980 4518 y FD(\000)2108 4457 y Fz(a)j(cl)r FA(\()p FB(T)2344 4471 y Fs(1)2384 4457 y FA(\))g Fz(d)p 2081 4498 436 4 v 2081 4581 a FA(1)d FD(\000)g Fz(cl)r FA(\()p FB(T)2402 4595 y Fs(1)2442 4581 y FA(\))p Fz(b)1407 4766 y FD(\024)1551 4705 y Fz(ad)k(cl)r FA(\()p FB(T)1835 4720 y Fr(d)1876 4705 y FA(\))p 1513 4746 437 4 v 1513 4829 a(1)c FD(\000)g Fz(cl)r FA(\()p FB(T)1834 4844 y Fr(d)1875 4829 y FA(\))p Fz(b)1407 5014 y FD(\024)1532 4953 y Fz(ad)j(cl)r FA(\()p FB(T)1815 4968 y Fr(d)1856 4953 y FA(\))p 1513 4994 398 4 v 1513 5077 a(1)d FD(\000)g Fz(cl)r FA(\()p FB(T)1834 5092 y Fr(d)1875 5077 y FA(\))1905 5349 y FF(38)p eop %%Page: 39 44 39 43 bop 300 391 a FF(So,)22 b(taking)g(the)g(ratio)h(of)f(the)g(two)g (gives)865 591 y Fz(cl)r FA(\()p FB(T)1030 606 y Fr(d)1071 591 y FB(T)1133 605 y Fs(1)1173 591 y Fz(s)p FA(\))e FD(\000)g Fz(cl)r FA(\()p Fz(s)p FA(\))p 865 631 678 4 v 916 714 a Fz(cl)r FA(\()p FB(T)1081 729 y Fr(d)1122 714 y Fz(s)p FA(\))g FD(\000)g Fz(cl)r FA(\()p Fz(s)p FA(\))1578 652 y FD(\025)1684 591 y Fz(ad)p FA(\()p Fz(cl)r FA(\()p FB(T)1979 606 y Fr(d)2021 591 y FB(T)2083 605 y Fs(1)2122 591 y FA(\))h FD(\000)f Fz(cl)r FA(\()p FB(T)g FD(\000)g FA(1\)\))p 1684 631 978 4 v 2029 692 a Fr(ad)j(cl)q Fs(\()p Fd(T)2253 704 y Fj(d)2290 692 y Fs(\))p 2029 716 289 4 v 2032 769 a(1)p Fn(\000)p Fr(cl)q Fs(\()p Fd(T)2250 781 y Fj(d)2286 769 y Fs(\))1578 949 y FA(=)1684 887 y(\()p Fz(cl)r FA(\()p FB(T)1884 902 y Fr(d)1926 887 y FB(T)1988 901 y Fs(1)2027 887 y FA(\))d FD(\000)g Fz(cl)r FA(\()p FB(T)h FD(\000)f FA(1\)\)\(1)i FD(\000)e Fz(cl)r FA(\()p FB(T)2923 902 y Fr(d)2964 887 y FA(\)\))p 1684 928 1351 4 v 2239 1011 a Fz(cl)r FA(\()p FB(T)2404 1026 y Fr(d)2445 1011 y FA(\))1578 1185 y(=)25 b Fz(K)32 b(>)25 b FA(0)e FF(,)g(a)g(constant.)p 3534 1446 4 62 v 3538 1388 55 4 v 3538 1446 V 3592 1446 4 62 v 300 1759 a Fo(3.5.1)99 b(Explicit)23 b(Construction)j(of)f(the)g(Limit)f(Set)300 1992 y FF(Now)-8 b(,)28 b(we)f(need)g(to)g(pr)n(ove)g FA(lim)1411 2006 y Fr(n)p Fn(!1)1614 1992 y Fz(\016)s FA(\(\003)p Fz(;)15 b FA(\000)1852 2006 y Fr(n)1900 1992 y FA(\))36 b(=)e(0)29 b FF(so)e(that,)i(in)f(conjunction)g(with)g (Lemma)300 2162 y(3.20,)k(we)d(may)h(establish)g(the)f(conver)n(gence)g (r)n(esult.)48 b(W)-8 b(e)29 b(shall)h(accomplish)h(this)e(thr)n(ough) 300 2331 y(the)22 b(following)h(steps:)414 2600 y(1.)45 b(Given)20 b(any)g Fz(\017)25 b(>)g FA(0)p FF(,)20 b(subdivide)f FA(\003)h FF(into)f Fz(k)k FF(intervals)d FA([)p Fz(l)2380 2614 y Fr(i)2408 2600 y Fz(;)15 b(r)2489 2614 y Fr(i)2518 2600 y FA(])20 b FF(wher)n(e)e(the)h(lengths)f Fz(r)3333 2614 y Fr(i)3370 2600 y FD(\000)8 b Fz(l)3476 2614 y Fr(i)3529 2600 y Fz(<)527 2770 y(\017=)p FA(2)24 b FF(for)e(all)i Fz(i)p FF(,)f(and)1191 2701 y Fq(S)1267 2796 y Fr(i)p Fn(2)p Fs([1:)p Fr(k)r Fs(])1474 2770 y FA([)p Fz(l)1526 2784 y Fr(i)1555 2770 y Fz(;)15 b(r)1636 2784 y Fr(i)1665 2770 y FA(])25 b(=)g(\003)p FF(.)414 3014 y(2.)45 b(For)17 b(each)h(interval)g FA([)p Fz(l)1262 3028 y Fr(i)1290 3014 y Fz(;)d(r)1371 3028 y Fr(i)1400 3014 y FA(])p FF(,)j(pick)g(the)e (midpoint)h Fz(p)2238 3028 y Fr(i)2292 3014 y FA(=)25 b(\()p Fz(r)2464 3028 y Fr(i)2493 3014 y FD(\000)q Fz(l)2592 3028 y Fr(i)2620 3014 y FA(\))p Fz(=)p FA(2)p FF(,)19 b(and)e(\002nd)g(a)h(backwar)n(d)527 3183 y(monotone)j(strategy)g (sequence)g FA(\()p Fz(s)1774 3139 y Fr(p)1810 3149 y Fj(i)1774 3195 y Fr(n)1840 3183 y FA(\))1875 3197 y Fr(n)p Fn(2)p Fl(N)2040 3183 y FF(such)h(that)g Fz(p)2475 3197 y Fr(i)2529 3183 y FA(=)j(lim)2751 3197 y Fr(n)p Fn(!1)2954 3183 y Fz(c)p FA([)p Fz(s)3061 3139 y Fr(p)3097 3149 y Fj(i)3061 3195 y Fr(n)3127 3183 y FA(])p FF(.)414 3427 y(3.)45 b(Since)28 b(ther)n(e)e(ar)n(e)i(only)e Fz(k)31 b FF(strategy)25 b(sequences,)h(ther)n(e)g(exists)g(an)i Fz(N)37 b FF(such)27 b(that)g(for)h(all)527 3596 y Fz(n)d(>)g(N)10 b FF(,)22 b FA(lim)958 3610 y Fr(m)p Fn(!1)1181 3596 y Fz(c)p FA([)p Fz(s)1288 3552 y Fr(p)1324 3562 y Fj(i)1288 3608 y Fr(m)1354 3596 y FA(])f FD(\000)f Fz(c)p FA([)p Fz(s)1598 3552 y Fr(p)1634 3562 y Fj(i)1598 3608 y Fr(n)1664 3596 y FA(])25 b Fz(<)g(\017=)p FA(2)f Fw(uniformly)f FF(in)h Fz(i)p FF(.)414 3840 y(4.)45 b(Thus)22 b(for)h(each)g(point)f Fz(\025)j FD(2)g FA(\003)p FF(,)d Fz(\025)k FD(2)e FA([)p Fz(l)1820 3854 y Fr(i)1849 3840 y Fz(;)15 b(r)1930 3854 y Fr(i)1959 3840 y FA(])23 b FF(for)f(some)g Fz(i)j FD(2)g FA([1)h(:)g Fz(k)s FA(])p FF(.)i(So,)22 b(for)g(all)i Fz(n)h(>)g(N)10 b FF(,)818 4101 y FD(j)p Fz(\025)20 b FD(\000)g Fz(c)p FA([)p Fz(s)1114 4064 y Fr(p)1150 4074 y Fj(i)1114 4124 y Fr(n)1181 4101 y FA(])p FD(j)25 b Fz(<)g FD(j)p Fz(\025)c FD(\000)f Fz(p)1588 4115 y Fr(i)1616 4101 y FD(j)g FA(+)g FD(j)p Fz(p)1823 4115 y Fr(i)1871 4101 y FD(\000)59 b FA(lim)1962 4156 y Fr(m)p Fn(!1)2181 4101 y Fz(c)p FA([)p Fz(s)2288 4064 y Fr(p)2324 4074 y Fj(i)2288 4124 y Fr(m)2355 4101 y FA(])p FD(j)20 b FA(+)g FD(j)54 b FA(lim)2556 4156 y Fr(m)p Fn(!1)2775 4101 y Fz(c)p FA([)p Fz(s)2882 4064 y Fr(p)2918 4074 y Fj(i)2882 4124 y Fr(m)2949 4101 y FA(])21 b FD(\000)f Fz(c)p FA([)p Fz(s)3193 4064 y Fr(p)3229 4074 y Fj(i)3193 4124 y Fr(n)3259 4101 y FA(])p FD(j)1256 4317 y Fz(<)1367 4255 y(\017)p 1362 4296 46 4 v 1362 4379 a FA(2)1438 4317 y(+)g(0)g(+)1700 4255 y Fz(\017)p 1695 4296 V 1695 4379 a FA(2)1256 4531 y Fz(<)25 b(\017)e(:)583 4829 y FF(Hence,)f(we)g(have)1249 5089 y FD(8)p Fz(\017)j(>)g FA(0)e FD(9)p Fz(N)34 b FD(2)25 b Fm(N)36 b FD(8)p Fz(n)24 b(>)h(N)33 b FA(\003)25 b FD(\022)g Fz(B)2444 5103 y Fr(\017)2477 5089 y FA(\(\000)2569 5103 y Fr(n)2616 5089 y FA(\))729 b FF(\(3.15\))1905 5349 y(39)p eop %%Page: 40 45 40 44 bop 300 391 a FF(and)23 b(thus)e FA(lim)802 405 y Fr(n)p Fn(!1)1005 391 y Fz(\016)s FA(\(\003)p Fz(;)15 b FA(\000)1243 405 y Fr(n)1291 391 y FA(\))26 b(=)f(0)e FF(as)g(r)n(equir)n(ed.)583 561 y(Consequently)-10 b(,)26 b(our)h(main)h(task)f(is)g(to)f(come)h(up)g(with)g(a)h(systematic)e (way)h(of)g(iden-)300 730 y(tifying)32 b(the)g(backwar)n(d)h(monotone)e (strategy)g(sequences)f FA(\()p Fz(s)2503 686 y Fr(p)2539 696 y Fj(i)2503 742 y Fr(n)2569 730 y FA(\))2604 744 y Fr(n)p Fn(2)p Fl(N)2779 730 y FF(satisfying)i(the)g(above)300 899 y(pr)n(operty)-10 b(.)300 1129 y Fp(Lemma)39 b(3.24.)52 b Fw(For)39 b(each)g Fz(\025)55 b FD(2)g FA(\003)p Fw(,)42 b(ther)n(e)f(exists)g(a)e(backward)h(monotone)f(strategy)j(sequence,) 300 1298 y FA(\()p Fz(s)378 1265 y Fr(\025)378 1320 y(n)425 1298 y FA(\))460 1312 y Fr(n)p Fn(2)p Fl(N)602 1298 y Fw(,)23 b(such)g(that)1678 1467 y FA(lim)1650 1522 y Fr(n)p Fn(!1)1849 1467 y Fz(c)p FA([)p Fz(s)1956 1430 y Fr(\025)1956 1490 y(n)2003 1467 y FA(])i(=)g Fz(\025)e(:)1130 b FF(\(3.16\))300 1708 y Fw(Pr)n(oof.)46 b FF(Let)22 b Fz(s)758 1676 y Fr(\025)758 1733 y Fs(0)828 1708 y FD(2)j Fm(S)970 1727 y Fs([0:0])1098 1708 y FA(\()p FD(T)e FA(\))g FF(be)g(such)f(that)1386 2051 y Fz(s)1429 2014 y Fr(\025)1429 2074 y Fs(0)1474 2051 y FA(\(0\))27 b(=)1712 1838 y Fq(8)1712 1920 y(>)1712 1947 y(<)1712 2110 y(>)1712 2138 y(:)1793 1968 y FB(T)1855 1983 y Fr(d)2009 1968 y FF(if)c Fz(cl)r FA(\()p FB(T)2253 1983 y Fr(d)2294 1968 y FA(\))j FD(\024)f Fz(\025)1793 2171 y FB(T)1855 2185 y Fs(1)2009 2171 y FF(otherwise,)3380 2051 y(\(3.17\))300 2393 y(and)e(let)f Fz(s)642 2360 y Fr(\025)642 2415 y(n)711 2393 y FF(be)h(de\002ned)e(r)n(ecursively)-10 b(,)22 b(for)g Fz(n)j FD(\025)g FA(1)p FF(,)e(as)1226 2726 y Fz(s)1269 2689 y Fr(\025)1269 2749 y(n)1341 2726 y FA(=)1437 2513 y Fq(8)1437 2594 y(>)1437 2622 y(<)1437 2785 y(>)1437 2813 y(:)1518 2642 y FB(T)1580 2657 y Fr(d)1620 2642 y Fz(s)1663 2610 y Fr(\025)1663 2667 y(n)p Fn(\000)p Fs(1)1800 2642 y FA(#)113 b FF(if)24 b Fz(cl)r FA(\()p FB(T)2234 2657 y Fr(d)2275 2642 y Fz(s)2318 2610 y Fr(\025)2318 2667 y(n)p Fn(\000)p Fs(1)2454 2642 y FA(\))i FD(\024)f Fz(\025)1518 2846 y FB(T)1580 2860 y Fs(1)1619 2846 y Fz(s)1662 2813 y Fr(\025)1662 2870 y(n)p Fn(\000)p Fs(1)1799 2846 y FA(#)114 b FF(otherwise.)3380 2726 y(\(3.18\))300 3068 y(Thus)22 b FA(\()p Fz(s)603 3035 y Fr(\025)603 3090 y(n)650 3068 y FA(\))h FF(is)f(a)i(backwar)n(d)f(monotone)e (sequence.)583 3237 y(Since)32 b Fz(c)p FA([)p Fz(s)933 3204 y Fr(\025)933 3259 y(n)981 3237 y FA(])42 b FD(\024)h Fz(c)p FA([)p Fz(s)1269 3204 y Fr(\025)1269 3261 y(n)p Fs(+1)1406 3237 y FA(])g FD(\024)f FA(1)32 b FF(holds)g(for)f(all)j Fz(n)p FF(,)f(the)f(sequence)e Fz(c)p FA([)p Fz(s)2975 3204 y Fr(\025)2975 3259 y(n)3022 3237 y FA(])i FF(conver)n(ges)f(as) 300 3406 y Fz(n)25 b FD(!)g(1)p FF(.)j(Thus,)21 b(we)h(need)g(to)g (show)f(that)i(the)e(limit)j(of)f(the)f(above)h(sequence)e(is)h(indeed) g Fz(\025)p FF(.)583 3576 y(First,)41 b(we)c(shall)i(establish)e(that)g Fz(cl)r FA(\()p Fz(s)1944 3543 y Fr(\025)1944 3598 y(n)1992 3576 y FA(\))53 b FD(\024)g Fz(\025)38 b FF(for)f(all)i Fz(n)f FF(by)f(induction.)73 b(For)37 b(the)300 3745 y(base)31 b(case)h(when)f Fz(n)41 b FA(=)h(0)p FF(,)34 b(note)c(that)i(if)g Fz(cl)r FA(\()p FB(T)1927 3760 y Fr(d)1968 3745 y FA(\))42 b FD(\024)g Fz(\025)p FF(,)33 b(then)e 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b(,)27 b(as)300 4253 y Fz(\025)e FD(2)g FA(\003)h(=)e([)q(lim)800 4267 y Fr(m)p Fn(!1)1023 4253 y Fz(c)p FA([)p FB(T)1149 4220 y Fr(m)1149 4278 y Fs(1)1216 4253 y FA(])p Fz(;)15 b FA(1])q FF(,)23 b(we)f(can)h (conclude)g(that)f Fz(cl)r FA(\()p Fz(s)2414 4220 y Fr(\025)2414 4278 y Fs(0)2460 4253 y FA(\))j(=)g(lim)2742 4267 y Fr(m)p Fn(!1)2965 4253 y Fz(c)p FA([)p FB(T)3091 4220 y Fr(m)3091 4278 y Fs(1)3158 4253 y FA(])h FD(\024)f Fz(\025)p FF(.)583 4423 y(For)17 b(the)g(inductive)h(step,)e(let)h(us)g(assume)g(the)g (hypothesis)e(that)i Fz(cl)r FA(\()p Fz(s)2954 4390 y Fr(\025)2954 4445 y(n)3001 4423 y FA(\))26 b FD(\024)f Fz(\025)p FF(,)18 b(and)g(look)300 4592 y(at)28 b Fz(cl)r FA(\()p Fz(s)549 4559 y Fr(\025)549 4616 y(n)p Fs(+1)686 4592 y FA(\))p FF(.)43 b(By)27 b(de\002nition,)g(if)i Fz(cl)r FA(\()p FB(T)1614 4607 y Fr(d)1655 4592 y Fz(s)1698 4559 y Fr(\025)1698 4614 y(n)1744 4592 y FA(\))35 b FD(\024)f Fz(\025)p FF(,)29 b(then)d Fz(s)2273 4559 y Fr(\025)2273 4616 y(n)p Fs(+1)2444 4592 y FA(=)35 b FB(T)2612 4607 y Fr(d)2652 4592 y Fz(s)2695 4559 y Fr(\025)2695 4614 y(n)2741 4592 y FF(.)43 b(Hence)27 b(for)g(this)g(case,)300 4761 y Fz(cl)r FA(\()p Fz(s)446 4728 y Fr(\025)446 4786 y(n)p Fs(+1)583 4761 y FA(\))h(=)g Fz(cl)r FA(\()p FB(T)910 4776 y Fr(d)951 4761 y Fz(s)994 4728 y Fr(\025)994 4784 y(n)1040 4761 y FA(\))g FD(\024)f Fz(\025)p FF(.)32 b(If,)24 b(on)f(the)h(other)e(hand,)i Fz(cl)r FA(\()p FB(T)2348 4776 y Fr(d)2389 4761 y Fz(s)2432 4728 y Fr(\025)2432 4784 y(n)2479 4761 y FA(\))j Fz(>)h(\025)p FF(,)c(then)f Fz(s)2986 4728 y Fr(\025)2986 4786 y(n)p Fs(+1)3150 4761 y FA(=)k FB(T)3310 4775 y Fs(1)3350 4761 y Fz(s)3393 4728 y Fr(\025)3393 4784 y(n)3439 4761 y FF(.)32 b(By)300 4931 y(the)24 b(de\002nition)h(of)g Fz(cl)r FA(\()p FD(\001)p FA(\))p FF(,)i Fz(cl)r FA(\()p Fz(s)1333 4898 y Fr(\025)1333 4955 y(n)p Fs(+1)1470 4931 y FA(\))j(=)g Fz(cl)r FA(\()p FB(T)1801 4945 y Fs(1)1841 4931 y Fz(s)1884 4898 y Fr(\025)1884 4953 y(n)1931 4931 y FA(\))g(=)g Fz(cl)r FA(\()p Fz(s)2243 4898 y Fr(\025)2243 4953 y(n)2290 4931 y FA(\))p FF(.)36 b(However)-7 b(,)24 b(as)h Fz(cl)r FA(\()p Fz(s)3056 4898 y Fr(\025)3056 4953 y(n)3104 4931 y FA(\))30 b FD(\024)g Fz(\025)25 b FF(by)g(the)300 5100 y(induction)d(hypothesis,)e(we)i (have)h Fz(cl)r FA(\()p Fz(s)1698 5067 y Fr(\025)1698 5124 y(n)p Fs(+1)1836 5100 y FA(\))i FD(\024)g Fz(\025)e FF(for)g(this)f(case)g(as)h(well.)1905 5349 y(40)p eop %%Page: 41 46 41 45 bop 583 391 a FF(Thus,)22 b(by)g(induction,)h(we)f(have)h(shown)e (that)i Fz(cl)r FA(\()p Fz(s)2366 358 y Fr(\025)2366 414 y(n)2413 391 y FA(\))i FD(\024)g Fz(\025)e FF(for)g(all)h Fz(n)p FF(.)583 561 y(Next,)e(notice)h(that,)f(for)h(any)g(strategy)e Fz(s)p FF(,)h Fz(c)p FA([)p FB(T)2166 528 y Fr(m)2166 588 y(d)2233 561 y Fz(s)p FA(])k FD(!)g FA(1)e FF(as)f Fz(m)i FD(!)h(1)p FF(,)d(and)g(so,)f Fz(cl)r FA(\()p FB(T)3455 528 y Fr(m)3455 588 y(d)3522 561 y Fz(s)p FA(\))300 730 y FF(conver)n(ges)f(to)h FA(1)i FF(as)e(well.)29 b(Thus,)21 b(for)i(each)g Fz(n)p FF(,)f(ther)n(e)f(exists)h(an)h Fz(M)35 b FA(=)25 b Fz(M)10 b FA(\()p Fz(n)p FA(\))23 b FF(such)f(that)1238 991 y Fz(cl)r FA(\()p FB(T)1403 953 y Fr(M)7 b Fn(\000)p Fs(1)1403 1020 y Fr(d)1572 991 y Fz(s)1615 953 y Fr(\025)1615 1013 y(n)1662 991 y FA(\))25 b FD(\024)g Fz(\025)e FF(and)g Fz(cl)r FA(\()p FB(T)2236 953 y Fr(M)2236 1014 y(d)2315 991 y Fz(s)2358 953 y Fr(\025)2358 1013 y(n)2405 991 y FA(\))i Fz(>)g(\025)e(:)300 1251 y FF(Hence)f(by)g(\(3.18\))r(,)g Fz(s)1010 1218 y Fr(\025)1010 1278 y(n)p Fs(+)p Fr(M)7 b Fn(\000)p Fs(1)1302 1251 y FA(=)25 b FB(T)1460 1204 y Fr(M)7 b Fn(\000)p Fs(1)p Fr(s)1658 1180 y Fj(\025)1658 1220 y(n)1460 1281 y Fr(d)1704 1251 y FF(.)583 1421 y(The)34 b(idea)g(her)n(e)g(is)h(to)e(gr)n(oup)h (the)f(triples)h(assigned)f(by)h(the)f(strategy)f Fz(s)3208 1388 y Fr(\025)3208 1443 y(n)3251 1451 y Fj(w)3339 1421 y FF(into)i Fz(w)300 1590 y FF(blocks)23 b(of)f(the)g(form:)1456 1760 y FD(\001)15 b(\001)g(\001)h FA([)p FB(T)1664 1775 y Fr(d)1720 1760 y FD(\001)f(\001)g(\001)h FB(T)1903 1775 y Fr(d)1602 1810 y Fq(|)p 1643 1810 89 11 v 89 w({z)p 1814 1810 V 89 w(})1616 1896 y Fr(m)1678 1904 y Fj(w)1745 1896 y Fc(copies)1959 1760 y FB(T)2021 1774 y Fs(1)2075 1760 y FD(\001)f(\001)g(\001)h FB(T)2258 1774 y Fs(1)1959 1809 y Fq(|)p 2000 1809 88 11 v 88 w({z)p 2170 1809 V 88 w(})1971 1896 y Fr(m)2033 1877 y Fi(0)2033 1912 y Fj(w)2100 1896 y Fc(copies)2298 1760 y FA(])f FD(\001)g(\001)g(\001)300 2056 y FF(wher)n(e)22 b Fz(n)628 2070 y Fr(w)710 2056 y FA(=)807 1988 y Fq(P)903 2014 y Fr(w)903 2083 y(i)p Fs(=1)1021 2056 y FA(\()p Fz(m)1136 2070 y Fr(w)1213 2056 y FA(+)f Fz(m)1385 2023 y Fn(0)1385 2079 y Fr(w)1441 2056 y FA(\))p FF(.)29 b(If)23 b(the)f(number)h(of)g(positions)f Fz(z)30 b FD(2)25 b FA([)p FD(\000)p Fz(n;)15 b FA(0])24 b FF(mapped)e(to)h FB(T)3560 2071 y Fr(d)300 2226 y FF(by)e(the)g (strategy)e Fz(s)958 2193 y Fr(\025)958 2248 y(n)1027 2226 y FF(appr)n(oaches)i(in\002nity)g(as)h Fz(n)j FD(!)g(1)p 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Fj(w)2033 3359 y Fs(1)2150 3333 y Fz(s)2193 3296 y Fr(\025)2193 3356 y(n)2236 3365 y Fj(w)q Fi(\000)p Fk(1)300 3594 y FF(wher)n(e)d Fz(n)628 3608 y Fr(w)709 3594 y FA(=)j Fz(m)885 3608 y Fr(w)962 3594 y FA(+)20 b Fz(m)1133 3561 y Fn(0)1133 3616 y Fr(w)1209 3594 y FA(+)g Fz(n)1355 3608 y Fr(w)r Fn(\000)p Fs(1)1502 3594 y FF(,)i(and)h Fz(m)1804 3608 y Fr(w)1885 3594 y Fz(>)i FA(0)p FF(,)e Fz(m)2152 3561 y Fn(0)2152 3616 y Fr(w)2234 3594 y Fz(>)i FA(0)e FF(for)f(all)j Fz(w)r FF(.)583 3763 y(If)e(this)f(subsequence)f(is)i(\002nite,)f(then)f(ther)n(e)h(is)h(a)g Fz(W)35 b FF(such)22 b(that)1627 4024 y Fz(s)1670 3986 y Fr(\025)1670 4046 y(n)1742 4024 y FA(=)j FB(T)1900 3981 y Fr(n)p Fn(\000)p Fr(n)2041 3992 y Fj(W)1900 4050 y Fs(1)2114 4024 y Fz(s)2157 3986 y Fr(\025)2157 4046 y(n)2200 4057 y Fj(W)300 4285 y FF(for)e(all)h Fz(n)g(>)h(n)790 4299 y Fr(W)871 4285 y FF(.)i(In)c(other)e(wor)n(ds,)g Fz(cl)r FA(\()p FB(T)1726 4300 y Fr(d)1767 4285 y FB(T)1829 4242 y Fr(n)1872 4219 y Fi(0)1894 4242 y Fn(\000)p Fr(n)1992 4253 y Fj(W)1829 4311 y Fs(1)2066 4285 y Fz(s)2109 4252 y Fr(\025)2109 4307 y(n)2152 4318 y Fj(W)2225 4285 y FA(\))k Fz(>)g(\025)e FF(for)g(all)h Fz(n)2772 4252 y Fn(0)2820 4285 y FD(\025)h Fz(n)2971 4299 y Fr(W)3051 4285 y FF(.)583 4454 y(However)-7 b(,)29 b(by)f(Pr)n(oposition)f(3.22,) j FA(lim)1966 4469 y Fr(n)2009 4450 y Fi(0)2031 4469 y Fn(!1)2192 4454 y Fz(cl)r FA(\()p FB(T)2357 4469 y Fr(d)2398 4454 y FB(T)2460 4411 y Fr(n)2503 4388 y Fi(0)2525 4411 y Fn(\000)p Fr(n)2623 4422 y Fj(W)2460 4480 y Fs(1)2696 4454 y Fz(s)2739 4421 y Fr(\025)2739 4476 y(n)2782 4487 y Fj(W)2855 4454 y FA(\))36 b(=)f Fz(cl)r FA(\()p Fz(s)3178 4421 y Fr(\025)3178 4476 y(n)3221 4487 y Fj(W)3295 4454 y FA(\))p FF(.)45 b(Thus)300 4623 y Fz(cl)r FA(\()p Fz(s)446 4590 y Fr(\025)446 4646 y(n)489 4657 y Fj(W)563 4623 y FA(\))50 b FD(\025)f Fz(\025)p FF(.)67 b(Y)-8 b(et)35 b Fz(cl)r FA(\()p Fz(s)1219 4590 y Fr(\025)1219 4646 y(n)1266 4623 y FA(\))50 b FD(\024)f Fz(\025)36 b FF(for)g(all)h 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847 y(1)1973 821 y Fz(s)2016 784 y Fr(\025)2016 844 y(n)2059 853 y Fj(w)q Fi(\000)p Fk(1)2191 821 y FA(\))k Fz(>)f(\025)1589 1016 y(cl)r FA(\()p FB(T)1754 1031 y Fr(d)1795 1016 y FB(T)1857 968 y Fr(m)1919 944 y Fi(0)1919 984 y Fj(w)1857 1042 y Fs(1)1973 1016 y Fz(s)2016 978 y Fr(\025)2016 1038 y(n)2059 1047 y Fj(w)q Fi(\000)p Fk(1)2191 1016 y FA(\))h FD(\024)f Fz(\025)300 1276 y FF(and)e(hence)802 1476 y Fz(\025)e FD(\000)e Fz(cl)r FA(\()p FB(T)1131 1491 y Fr(d)1172 1476 y FB(T)1234 1429 y Fr(m)1296 1405 y Fi(0)1296 1445 y Fj(w)1234 1502 y Fs(1)1351 1476 y Fz(s)1394 1443 y Fr(\025)1394 1499 y(n)1437 1508 y Fj(w)q Fi(\000)p Fk(1)1569 1476 y FA(\))p 802 1530 802 4 v 943 1613 a Fz(\025)h FD(\000)g Fz(cl)r FA(\()p Fz(s)1253 1587 y Fr(\025)1253 1635 y(n)1296 1644 y Fj(w)q Fi(\000)p Fk(1)1428 1613 y FA(\))1639 1551 y(=)25 b(1)c FD(\000)1902 1476 y Fz(cl)r FA(\()p FB(T)2067 1491 y Fr(d)2108 1476 y FB(T)2170 1429 y Fr(m)2232 1405 y Fi(0)2232 1445 y Fj(w)2170 1502 y Fs(1)2286 1476 y Fz(s)2329 1443 y Fr(\025)2329 1499 y(n)2372 1508 y Fj(w)q Fi(\000)p Fk(1)2504 1476 y FA(\))g FD(\000)f Fz(cl)r FA(\()p Fz(s)2797 1443 y Fr(\025)2797 1499 y(n)2840 1508 y Fj(w)q Fi(\000)p Fk(1)2972 1476 y FA(\))p 1902 1530 1106 4 v 2194 1613 a Fz(\025)g FD(\000)g Fz(cl)r FA(\()p Fz(s)2504 1587 y Fr(\025)2504 1635 y(n)2547 1644 y Fj(w)q Fi(\000)p Fk(1)2680 1613 y FA(\))1639 1858 y FD(\024)25 b FA(1)c FD(\000)1947 1784 y Fz(cl)r FA(\()p FB(T)2112 1799 y Fr(d)2153 1784 y FB(T)2215 1736 y Fr(m)2277 1712 y Fi(0)2277 1752 y Fj(w)2215 1810 y Fs(1)2331 1784 y Fz(s)2374 1751 y Fr(\025)2374 1806 y(n)2417 1815 y Fj(w)q Fi(\000)p Fk(1)2549 1784 y FA(\))g FD(\000)f Fz(cl)r FA(\()p Fz(s)2842 1751 y Fr(\025)2842 1806 y(n)2885 1815 y Fj(w)q Fi(\000)p Fk(1)3017 1784 y FA(\))p 1902 1837 1196 4 v 1902 1942 a Fz(cl)r FA(\()p FB(T)2067 1957 y Fr(d)2108 1942 y FB(T)2170 1894 y Fr(m)2232 1876 y Fi(0)2232 1911 y Fj(w)2282 1894 y Fn(\000)p Fs(1)2170 1968 y(1)2376 1942 y Fz(s)2419 1916 y Fr(\025)2419 1965 y(n)2462 1974 y 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y(The)k(fundamental)h(r)n(esult)f(pr)n(oven)g(in)h(this)f(thesis)f(is)i (that,)g(given)f(an)h(internal)g(state)300 4108 y(space)i(that)f (contains)h(only)g(one)f(distinguished)f(state,)i(and)g(a)g(sequence)f (of)h(sets)e(of)i(strate-)300 4278 y(gies)21 b(of)h(incr)n(easing)h (length,)e(the)g(characteristic)i(tuple)f(sets)e(of)i(the)f(corr)n (esponding)f(random)300 4447 y(set-walks)26 b(conver)n(ge,)h(on)g(a)h (term-by-term)e(basis,)j(in)f(the)e(Hausdorf)n(f)h(metric)g(on)g(sets.) 40 b(W)-8 b(e)300 4616 y(pr)n(oved)19 b(this)g(Main)i(Theor)n(em)e(in)h (several)g(parts,)f(for)h(dif)n(fer)n(ent)f(compositions)g(of)h(the)f (transi-)300 4786 y(tion)i(set.)26 b(For)20 b(the)g(case)g(with)h(pur)n (ely)f(left-leaning)i(transition)e(triples,)g(we)h(utilized)g(the)f (idea)300 4955 y(of)h(bounding)f(the)g(sets)f FA(\000)1190 4969 y Fr(n)1258 4955 y FF(by)i(appr)n(oximations)g Fz(L)2098 4969 y Fr(n)2165 4955 y FF(and)g Fz(I)2380 4969 y Fr(n)2427 4955 y FF(,)g(and)g(pr)n(oceeded)e(to)h(pr)n(ove)g(that)1905 5349 y(48)p eop %%Page: 49 54 49 53 bop 300 391 a Fz(I)340 405 y Fr(n)413 391 y FF(and)26 b Fz(L)655 405 y Fr(n)728 391 y FF(conver)n(ge)f(to)h(the)f(same)h (limit,)i(entailing)f(the)e(conver)n(gence)g(of)h FA(\000)3076 405 y Fr(n)3149 391 y FF(itself.)38 b(W)-5 b(ith)300 561 y(mixed)32 b(transition)g(sets,)g(we)g(speci\002ed)e(a)j(scheme)e (\226)i(a)f(gr)n(eedy)f(algorithm)h(\226)h(with)f(which)300 730 y(we)23 b(can)h(systematically)f(identify)g(that)g(each)h(point)e Fz(\025)i FF(of)f(the)f(candidate)i(limit)g(set)f FA(\003)g FF(can)h(be)300 899 y(appr)n(oached)29 b(by)h(a)g(sequence)d(of)j (points)e FA(\()p Fz(c)p FA([)p Fz(s)1942 866 y Fr(\025)1942 922 y(n)1990 899 y FA(]\))i FF(taken)f(fr)n(om)h(the)f(sets)f FA(\000)2949 913 y Fr(n)2995 899 y FF(.)49 b(For)29 b(the)f(other)300 1069 y(cases,)c(we)f(employed)g(r)n(esults)g(fr)n(om)h(the)f(gambler)7 b('s)25 b(r)o(uin)f(pr)n(oblem)g(to)g(derive)f(the)h(desir)n(ed)300 1238 y(r)n(esult.)30 b(Finally)-10 b(,)24 b(we)f(consolidated)g(these)f (subpr)n(oofs)g(and)h(established)g(the)g(validity)h(of)g(the)300 1408 y(Main)g(Theor)n(em)d(in)i(its)g(most)e(general)i(form.)300 1781 y Ft(4.1)119 b(Futur)n(e)31 b(W)-11 b(ork)300 2048 y FF(While)21 b(the)g(Main)h(Theor)n(em)e(is)h(an)h(important)e(r)n (esult)h(concerning)f(the)h(long-r)o(un)g(behaviour)300 2218 y(of)i(random)g(set-walks,)f(it)i(is)f(limited)h(in)f(scope,)f (since)h(it)h(r)n(elies)f(on)g(the)f(singleton)g(natur)n(e)h(of)300 2387 y(the)g(internal)i(state)e(space.)31 b(In)24 b(particular)-7 b(,)25 b(it)g(does)d(not)h(solve)h(the)f(pr)n(oblem)h(of)g (establishing)300 2556 y(the)i(r)n(egularity)g(of)h(the)f(language)h (class)g Fp(2NPF)-5 b(A-unary)p FF(,)26 b(as)h(described)f(in)h(the)f (Motivation)300 2726 y(section.)h(W)-8 b(e)20 b(conjectur)n(e)h(that)g (a)h(generalization)f(of)h(the)e(Main)j(Theor)n(em)d(to)h(arbitrary)g (\002nite)300 2895 y(internal)i(state)f(spaces)f(is)i(also)g(valid.)300 3139 y Fp(Conjecture)k(4.1.)45 b Fw(For)27 b(any)g(\002nite)g(internal) h(state)h(space)f Fz(E)5 b Fw(,)28 b(and)f(any)h(non-empty)g (transition)g(set)300 3309 y FD(T)23 b Fw(,)c(the)i(sequence)e(of)h (characteristic)j(tuple)d(sets)h FA(\(\010)1943 3323 y Fr(n)1990 3309 y FA(\()p FD(T)i FA(\)\))d Fw(conver)n(ges)i (componentwise)e(in)f(the)i(Haus-)300 3478 y(dorff)j(metric.)583 3722 y FF(During)j(pr)n(eliminary)g(investigation,)g(it)g(was)f(noted)f (that)h(the)g(sequence)f(of)h(sets)f(of)300 3891 y(matrices)g FA(\000)722 3905 y Fr(n)793 3891 y FF(appr)n(oaches)f(a)h(\223limit)h (set\224)e(which)h(looks)e(macr)n(oscopically)j(like)f(a)g(collection) 300 4061 y(of)33 b(matrix)g(pencils.)58 b(Mor)n(eover)-7 b(,)34 b(the)d(mapping)i Fz(c)p FA([)p Fm(S)2196 4079 y Fs([1:)p Fr(n)o Fs(])2331 4061 y FA(\()p FD(T)23 b FA(\)])44 b FD(7!)g Fz(c)p FA([)p Fm(S)2799 4079 y Fs([1:)p Fr(n)o Fs(])2934 4061 y FA(\()p FD(T)23 b FA(\))28 b FD(\001)g Fz(s)3201 4028 y Fn(0)3224 4061 y FA(])33 b FF(appears)300 4230 y(to)25 b(be)h(a)h(combination)g(of)f(r)n(otation,) g(translation,)g(and)g(scaling)h(applied)f(non-uniformly)g(to)300 4400 y(the)j(points)g(in)h FA(\000)904 4414 y Fr(n)989 4400 y FA(=)38 b Fz(c)p FA([)p Fm(S)1219 4418 y Fs([1:)p Fr(n)p Fs(])1354 4400 y FA(\()p FD(T)23 b FA(\)])p FF(.)50 b(Hence,)31 b(it)f(may)g(be)f(fr)o(uitful)i(to)e(develop)g(an)h (analogue)300 4569 y(of)d(the)g(method)f(pr)n(esented)f(in)i(Section)g (3.3,)i(bounding)e(each)h(pencil)f(individually)i(by)f(ap-)300 4738 y(pr)n(oximating)36 b(sets,)i(and)e(then)f(pr)n(oving)g(that)h (the)f(sequences)f(of)i(these)e(appr)n(oximations)300 4908 y(conver)n(ge.)1905 5349 y(49)p eop %%Page: 50 55 50 54 bop 2408 997 a FC(Bibliography)345 1463 y FF([1])47 b(Mar)n(c)37 b(Artzr)n(ouni,)j Fw(A)c(theor)n(em)j(on)d(pr)n(oducts)j (of)e(matrices)p FF(,)k(Linear)36 b(Algebra)g(Appl.)g Fp(49)497 1587 y FF(\(1983\),)25 b(153\226159.)345 1786 y([2])p 497 1786 273 4 v 335 w(,)e Fw(On)f(the)i(conver)n(gence)i(of)e (in\002nite)f(pr)n(oducts)i(of)f(matrices)p FF(,)h(Linear)e(Algebra)g (Appl.)497 1910 y Fp(74)h FF(\(1986\),)g(1)-5 b(1\22621.)345 2109 y([3])p 497 2109 V 335 w(,)38 b Fw(On)c(the)i(gr)n(owth)h(of)f (in\002nite)f(pr)n(oducts)i(of)f(slowly)g(varying)h(primitive)h (matrices)p FF(,)497 2233 y(Linear)22 b(Algebra)h(Appl.)f Fp(145)i FF(\(1991\),)h(33\22657.)345 2432 y([4])p 497 2432 V 335 w(,)18 b Fw(On)g(the)h(dynamics)g(of)g(the)g(linear)g(pr)n (ocess)i Fz(Y)f FA(\()p Fz(k)s FA(\))26 b(=)f Fz(A)p FA(\()p Fz(k)s FA(\))p Fz(Y)c FA(\()p Fz(k)6 b FD(\000)s FA(1\))19 b Fw(with)f(irr)n(educible)497 2556 y(matrices)24 b Fz(A)p FA(\()p Fz(k)s FA(\))p FF(,)g(SIAM)e(J.)g(Matrix)i(Anal.)f (Appl.)e Fp(17)j FF(\(1996\),)h(no.)d(4,)h(822\226833.)345 2755 y([5])47 b(Michael)27 b(N.)e(Barber)h(and)g(B.)f(W)-8 b(.)25 b(Ninham,)i Fw(Random)f(and)g(r)n(estricted)k(walks.)c(Theory)i (and)497 2879 y(applications)p FF(,)d(Gor)n(don)d(and)h(Br)n(each,)g (New)e(Y)-8 b(ork-London-Paris,)20 b(1970.)345 3078 y([6])47 b(Abraham)31 b(Berman)e(and)h(Robert)e(J.)i(Plemmons,)h Fw(Nonnegative)g(matrices)h(in)d(the)i(math-)497 3202 y(ematical)46 b(sciences)p FF(,)51 b(Society)44 b(for)g(Industrial)g (and)g(Applied)g(Mathematics)h(\(SIAM\),)497 3327 y(Philadelphia,)24 b(P)-8 b(A,)21 b(1994,)k(Revised)c(r)n(eprint)h(of)h(the)f(1979)i (original.)345 3526 y([7])47 b(S.)33 b(Chatterjee)g(and)h(E.)f(Seneta,) i Fw(T)-10 b(owards)35 b(consensus:)51 b(some)35 b(conver)n(gence)h (theor)n(ems)g(on)497 3650 y(r)n(epeated)26 b(averaging)p FF(,)f(J.)e(Appl.)f(Pr)n(obability)i Fp(14)f FF(\(1977\),)i(no.)d(1,)h (89\22697.)345 3849 y([8])47 b(Joel)29 b(E.)g(Cohen,)i(John)e(Hajnal,)k (and)c(Charles)h(M.)g(Newman,)g Fw(Appr)n(oaching)j(consensus)497 3973 y(can)26 b(be)h(delicate)i(when)e(positions)i(harden)p FF(,)g(Stochastic)d(Pr)n(ocess.)f(Appl.)h Fp(22)h FF(\(1986\),)j(no.)c (2,)497 4097 y(315\226322.)345 4296 y([9])47 b(Harry)35 b(Cohn)g(\(ed.\),)j Fw(Doeblin)d(and)h(modern)g(pr)n(obability)p FF(,)42 b(Pr)n(ovidence,)c(RI,)c(American)497 4420 y(Mathematical)24 b(Society)-10 b(,)22 b(1993.)300 4619 y([10])47 b(Anne)34 b(Condon,)j(Lisa)d(Hellerstein,)j(Samuel)e(Pottle,)h(and)f(A)-8 b(vi)35 b(W)-5 b(igderson,)36 b Fw(On)e(the)497 4743 y(power)22 b(of)g(\002nite)g(automata)g(with)g(both)g(nondeterministic) h(and)f(pr)n(obabilistic)j(states)p FF(,)e(SIAM)d(J.)497 4868 y(Comput.)i Fp(27)h FF(\(1998\),)i(no.)d(3,)h(739\226762)j (\(electr)n(onic\).)305 5066 y([1)-5 b(1])47 b(J.)23 b(L.)f(Doob,)g Fw(Stochastic)j(pr)n(ocesses)p FF(,)g(John)e(W)-5 b(iley)23 b(&)f(Sons)g(Inc.,)g(New)f(Y)-8 b(ork,)21 b(1953.)1905 5349 y(50)p eop %%Page: 51 56 51 55 bop 300 391 a FF([12])47 b(Jewgeni)33 b(H.)g(Dshalalow)i (\(ed.\),)g Fw(Advances)g(in)f(queueing.)f(Theory)m(,)38 b(methods,)g(and)c(open)497 516 y(pr)n(oblems)p FF(,)25 b(CRC)e(Pr)n(ess,)e(Boca)i(Raton,)f(FL,)f(1995.)300 714 y([13])47 b(Ioannis)29 b(I.)f(Ger)n(ontidis,)i Fw(Periodic)h(str)n(ong) g(er)n(godicity)h(in)d(nonhomogeneous)i(Markov)g(sys-)497 839 y(tems)p FF(,)23 b(J.)g(Appl.)f(Pr)n(obab.)h Fp(28)h FF(\(1991\),)g(no.)e(1,)h(58\22673.)300 1038 y([14])47 b(M.)37 b(Girault,)42 b Fw(Stochastic)e(pr)n(ocesses)p FF(,)1823 1016 y(\250)1802 1038 y(Okonometrie)c(und)h (Unternehmensforschung,)497 1162 y(III,)21 b(Springer)n(-V)-10 b(erlag)22 b(New)f(Y)-8 b(ork,)21 b(Inc.,)h(New)g(Y)-8 b(ork,)21 b(1966.)300 1361 y([15])47 b(Roe)c(Goodman,)50 b Fw(Intr)n(oduction)c(to)f(stochastic)i(models)p FF(,)52 b(The)43 b(Benjamin/Cummings)497 1485 y(Publishing)23 b(Co.)f(Inc.,)h(Menlo)f(Park,)g(CA,)h(1988.)300 1684 y([16])47 b(G.)33 b(R.)f(Grimmett)g(and)h(D.)g(R.)f(Stirzaker)-7 b(,)34 b Fw(Pr)n(obability)i(and)d(random)g(pr)n(ocesses)p FF(,)38 b(second)497 1808 y(ed.,)21 b(Oxfor)n(d)i(Science)g (Publications,)g(1993.)300 2007 y([17])47 b(Allan)d(Gut,)j Fw(Stopped)d(random)g(walks.)f(Limit)g(theor)n(ems)i(and)e (applications)p FF(,)51 b(Springer)n(-)497 2131 y(V)-10 b(erlag,)22 b(New)f(Y)-8 b(ork,)21 b(1988.)300 2330 y([18])47 b(J.)33 b(Hajnal,)k Fw(The)c(er)n(godic)j(pr)n(operties)h(of)d (non-homogeneous)g(\002nite)g(Markov)h(chains)p FF(,)h(Pr)n(oc.)497 2454 y(Cambridge)23 b(Philos.)f(Soc.)h Fp(52)g FF(\(1956\),)i (67\22677.)300 2653 y([19])p 497 2653 273 4 v 335 w(,)39 b Fw(W)n(eak)d(er)n(godicity)k(in)35 b(non-homogeneous)j(Markov)f (chains)p FF(,)j(Pr)n(oc.)c(Cambridge)497 2777 y(Philos.)22 b(Soc.)g Fp(54)i FF(\(1958\),)h(233\226246.)300 2976 y([20])p 497 2976 V 335 w(,)36 b Fw(On)e(pr)n(oducts)i(of)f (non-negative)h(matrices)p FF(,)i(Math.)c(Pr)n(oc.)g(Cambridge)g (Philos.)497 3101 y(Soc.)22 b Fp(79)i FF(\(1976\),)h(no.)d(3,)g (521\226530.)300 3299 y([21])p 497 3299 V 335 w(,)81 b Fw(Shuf\003ing)69 b(with)h(two)g(matrices)p FF(,)82 b(Doeblin)70 b(and)g(modern)e(pr)n(obability)497 3424 y(\(Blaubeur)n(en,)22 b(1991\))j(\(Pr)n(ovidence,)d(RI\),)g(Amer)-7 b(.)23 b(Math.)f(Soc.,)g(1993,)i(pp.)e(271\226287.)300 3623 y([22])47 b(Darald)24 b(J.)f(Hart\002el,)f Fw(Markov)i(set-chains) p FF(,)h(Springer)n(-V)-10 b(erlag,)21 b(Berlin,)h(1998.)300 3821 y([23])47 b(John)j(E.)g(Hopcr)n(oft)f(and)i(Jef)n(fr)n(ey)f(D.)h (Ullman,)58 b Fw(Intr)n(oduction)52 b(to)g(automata)f(theory)m(,)497 3946 y(languages,)61 b(and)53 b(computation)p FF(,)60 b(Addison-W)-8 b(esley)49 b(Series)i(in)i(Computer)e(Science,)497 4070 y(Addison-W)-8 b(esley)19 b(Publishing)k(Co.,)g(Reading,)e(Mass.,) h(1979.)300 4269 y([24])47 b(Dean)29 b(L.)g(Isaacson)g(and)g(Richar)n (d)h(W)-8 b(.)29 b(Madsen,)g Fw(Markov)i(chains.)f(Theory)h(and)f (applica-)497 4393 y(tions)p FF(,)d(Robert)d(E.)h(Krieger)g(Publishing) h(Co.)f(Inc.,)h(Melbourne,)f(Fla.,)i(1985,)h(Reprint)c(of)497 4517 y(the)e(1976)i(original.)300 4716 y([25])47 b(J)8 b(\257)-38 b(anis)31 b(Ka)846 4780 y FA(')832 4716 y FF(neps,)g Fw(Regularity)h(of)f(one-letter)h(languages)g(acceptable)h (by)e FA(2)p Fw(-way)g(\002nite)g(pr)n(oba-)497 4840 y(bilistic)21 b(automata)p FF(,)f(Fundamentals)f(of)f(computation)h (theory)e(\(Gosen,)i(1991\),)i(Springer)-7 b(,)497 4965 y(Berlin,)22 b(1991,)i(pp.)e(287\226296.)1905 5349 y(51)p eop %%Page: 52 57 52 56 bop 300 391 a FF([26])47 b(John)22 b(G.)h(Kemeny)-10 b(,)21 b(J.)h(Laurie)g(Snell,)g(and)h(Anthony)e(W)-8 b(.)21 b(Knapp,)h Fw(Denumerable)h(Markov)497 516 y(chains)p FF(,)55 b(second)46 b(ed.,)53 b(Springer)n(-V)-10 b(erlag,)52 b(New)47 b(Y)-8 b(ork,)52 b(1976,)j(W)-5 b(ith)48 b(a)g(chapter)f(on) 497 640 y(Markov)31 b(random)f(\002elds,)i(by)f(David)h(Grif)n(feath,)h (Graduate)e(T)-8 b(exts)29 b(in)i(Mathematics,)497 764 y(No.)22 b(40.)300 960 y([27])47 b(Andr)8 b(\264)-38 b(as)25 b(Kr)8 b(\264)-38 b(amli,)27 b(N)8 b(\264)-38 b(andor)25 b(Sim)8 b(\264)-38 b(anyi,)26 b(and)g(Domokos)e(Sz)8 b(\264)-38 b(asz,)27 b Fw(Random)e(walks)i(with)f(in-)497 1084 y(ternal)c(degr)n(ees)i(of)d(fr)n(eedom.)i(III.)e(Stationary)i(pr) n(obabilities)p FF(,)i(Pr)n(obab.)c(Theory)f(Relat.)g(Fields)497 1208 y Fp(72)k FF(\(1986\),)g(no.)e(4,)h(603\226617.)300 1404 y([28])47 b(Andr)8 b(\264)-38 b(as)23 b(Kr)8 b(\264)-38 b(amli)26 b(and)f(Domokos)e(Sz)8 b(\264)-38 b(asz,)25 b Fw(Random)g(walks)g(with)g(internal)g(degr)n(ees)j(of)d(fr)n(ee-)497 1528 y(dom.)e(I.)g(Local)g(limit)g(theor)n(ems)p FF(,)j(Z.)c(W)-8 b(ahrsch.)22 b(V)-10 b(erw)i(.)21 b(Gebiete)h Fp(63)h FF(\(1983\),)i(no.)d(1,)h(85\22695.)300 1724 y([29])p 497 1724 273 4 v 335 w(,)29 b Fw(Random)g(walks)g(with)g(internal)g (degr)n(ees)i(of)e(fr)n(eedom.)h(II.)f(First-hitting)i(pr)n(obabili-) 497 1848 y(ties)p FF(,)24 b(Z.)e(W)-8 b(ahrsch.)22 b(V)-10 b(erw)i(.)21 b(Gebiete)h Fp(68)h FF(\(1984\),)i(no.)d(1,)h(53\22664.) 300 2044 y([30])47 b(F)-8 b(.)20 b(Thomson)g(Leighton)g(and)h(Ronald)g (L.)g(Rivest,)g Fw(The)h(Markov)h(chain)f(tr)n(ee)h(theor)n(em)p FF(,)h(MIT)497 2168 y(Laboratory)19 b(for)g(Computer)g(Science)h (TM-249,)h(Massachusetts)d(Institute)g(of)i(T)-8 b(echnol-)497 2292 y(ogy)e(,)21 b(Cambridge,)i(Mass.,)f(November)f(1983.)300 2488 y([31])47 b(G)12 b(\250)-42 b(unter)21 b(Mayer)-7 b(,)22 b Fw(On)f(the)i(conver)n(gence)h(of)f(the)g(Neumann)d(series)k (in)e(interval)i(analysis)p FF(,)g(Lin-)497 2612 y(ear)f(Algebra)f (Appl.)g Fp(65)i FF(\(1985\),)h(63\22670.)300 2808 y([32])47 b(Michel)26 b(Morvan,)f(Christoph)g(Meinel,)g(and)g(Daniel)h(Kr)n(ob)g (\(eds.\),)e Fw(STACS)f(98)p FF(,)j(Berlin,)497 2932 y(Springer)n(-V)-10 b(erlag,)21 b(1998.)300 3128 y([33])47 b(V)-12 b(.)34 b(I.)g(Romanovsky)f(and)h(E.)g(Seneta,)i Fw(Discr)n(ete)g(Markov)g(chains)p FF(,)i(W)-8 b(olters-Noor)n(dhof)n (f)497 3252 y(Publishing,)32 b(Gr)n(oningen,)g(The)e(Netherlands,)g (1970,)k(T)-8 b(ranslated)30 b(fr)n(om)h(the)f(Russian)497 3376 y(by)22 b(E.)g(Seneta.)300 3572 y([34])47 b(Michael)30 b(Saks)f(and)g(Anne)f(Condon,)i Fw(A)f(limit)h(theor)n(em)h(for)f(sets) h(of)f(stochastic)i(matrices)p FF(,)497 3696 y(manuscript,)22 b(12)i(pages.)300 3892 y([35])47 b(Arto)27 b(Salomaa)i(and)f(Matti)g (Soittola,)h Fw(Automata-theor)n(etic)j(aspects)e(of)f(formal)g(power)g (se-)497 4016 y(ries)p FF(,)23 b(T)-8 b(exts)21 b(and)h(Monographs)e (in)j(Computer)e(Science,)h(Springer)n(-V)-10 b(erlag,)21 b(New)g(Y)-8 b(ork,)497 4140 y(1978.)300 4336 y([36])47 b(E.)17 b(Seneta,)h Fw(Non-negative)i(matrices)g(and)f(Markov)h(chains) p FF(,)g(second)d(ed.,)h(Springer)n(-V)-10 b(erlag,)497 4460 y(New)22 b(Y)-8 b(ork,)20 b(1981.)300 4656 y([37])47 b(Y)-8 b(a.)18 b(G.)h(Sinai,)i Fw(Random)e(walks)h(and)f(some)h(pr)n (oblems)h(concerning)e(lor)n(entz)i(gas)p FF(,)g(Pr)n(oceedings)497 4780 y(of)i(the)e(Kyoto)h(Confer)n(ence,)f(1981,)k(pp.)c(6\22617.)300 4976 y([38])47 b(Frank)20 b(Spitzer)-7 b(,)21 b Fw(Principles)i(of)e (random)h(walk)p FF(,)g(The)e(University)g(Series)g(in)i(Higher)e (Math-)497 5100 y(ematics,)j(D.)f(V)-8 b(an)23 b(Nostrand)e(Co.,)h (Inc.,)g(Princeton,)g(N.J.-T)-8 b(or)n(onto-London,)18 b(1964.)1905 5349 y(52)p eop %%Page: 53 58 53 57 bop 300 391 a FF([39])47 b(Geor)n(ge)32 b(H.)h(W)-8 b(eiss,)35 b Fw(Aspects)f(and)g(applications)j(of)d(the)g(random)h (walk)p FF(,)h(North-Holland)497 516 y(Publishing)23 b(Co.,)f(Amster)n(dam,)g(1994.)1905 5349 y(53)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF