%!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: jacks4.dvi %%CreationDate: Sat Oct 30 17:46:20 1999 %%Pages: 13 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: D:\EMTEX\BIN\dvips32.exe -pj=tmp.mfj -K jacks4 %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.10.30:1645 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true 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D<15F0141FA214011400AFEB1FC0EBFFF03801F03C3807C006380F8003EB0001 001E13005AA2127C1278A212F8A81278127C123CA26C130114036C13073807C00E3903F0 38F83A00FFF0FF80EB3F8021327EB125>II<120FEA1F80A4EA0F00C7FCAB EA0780127FA2120F1207B3A6EA0FC0EAFFF8A20D307EAF12>105 D107 DI<380781FE39FF87FF8090388E07C0390F9803E03807B0019038E0 00F05BA35BB3486C487E3AFFFC1FFF80A2211F7E9E25>110 DI<380783E038FF8FF8EB9C7CEA0FB8EA07F0EBE03814005BA35BB2487EEAFFFEA2161F 7E9E19>114 D<3801FC10380FFF30381E03F0EA38004813705A1430A37E6C1300127EEA 3FF0EA1FFE380FFF80000313E038003FF0EB03F8EB007800C0133CA2141C7EA27E14186C 13386C137038EF01E038C3FFC03880FE00161F7E9E1A>I<13C0A51201A31203A2120712 0F121FB512E0A23803C000B01430A83801E060A23800F0C0EB7F80EB1F00142C7FAB19> I<3BFFF07FF81FFCA23B0F8007C007F00203EB03C00007ED01801407A2D803C09038E003 00A2140CD801E0EBF006A2EC1870D800F0EB780CA2EC30380178EB3C18A2EC601C013CEB 1E30A2ECC00E011EEB0F60A290381F800716E0010F5CA2EC00036D5CA22E1F7F9E30> 119 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 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y(on)g(the)f(lo)q(cation)i(of)f (a)f(cell)h(\(non{homogeneit)o(y\).)12 2378 y(The)j(pap)q(er)i(is)f (organized)h(as)f(follo)o(ws:)32 b(in)22 b(section)f(2)g(the)g(mo)q (del)g(of)g(zero{range)g(in)o(teraction)h(is)-59 2437 y(in)o(tro)q(duced,)15 b(necessary)g(de\014nitions)i(are)e(made)g(and)g (the)g(existence)g(theorem)f(is)i(pro)o(v)o(ed.)k(The)15 b(results)-59 2495 y(of)k(section)g(3)g(yield)g(in)o(v)m(arian)o(t)h (measures)e(and)h(the)f(c)o(haracter)g(of)h(transience.)28 b(Section)19 b(4)g(is)g(dev)o(oted)-59 2553 y(to)e(the)f(in)o(v)o (estigation)i(of)e(sp)q(ecial)i(case)e Fo(\015)h(<)d Fn(1)p Fw(.)p -59 2593 780 2 v -8 2620 a Fk(\003)10 2636 y Fj(Mosco)o(w)f(State)h(Univ)o(ersit)o(y)m(,)g(Russia.)k(E-mail:)h (dima@vvv.srcc.msu.su)-6 2667 y Fk(y)10 2683 y Fj(F)m(riedric)o(h-Sc)o (hil)q(le)q(r)c(Univ)o(ersit\177)-19 b(at)15 b(Jena,)e(Deutsc)o(hland.) 19 b(E-mail:)g(seu@minet.uni-jena.de)904 2936 y Fi(1)p eop %%Page: 2 2 2 1 bop -59 140 a Fp(2)67 b(Mo)r(del:)29 b(Existence)23 b(and)g(Monotonicit)n(y)-59 248 y Fw(Consider)d(the)f(follo)o(wing)j (non{homogeneous)f(zero{range)f(in)o(teraction)g(mo)q(del)g(at)g(Bose)f ({)g(Einstein)-59 306 y(sp)q(eeds:)36 b(a)24 b(n)o(um)o(b)q(er)f(of)h (indistinguisha)q(ble)j(particles)d(is)h(lo)q(cated)g(in)f(a)g(coun)o (table)g(set)f(of)h(sites)h Fo(J)5 b Fw(.)-59 364 y(T)l(ransitions)23 b(of)e(particles)h(at)g(a)f(site)g Fo(i)g Fn(2)h Fo(J)j Fw(are)c(made)g(after)f(random)i(p)q(erio)q(ds)g(of)f(time)g Fo(\034)27 b Fw(|)20 b(i.)36 b(i.)-59 422 y(exp)q(onen)o(tially)21 b(d.)31 b(r.)f(v.)h(with)20 b(parameter)f Fo(\015)826 429 y Fm(i)841 422 y Fw(:)27 b(if)20 b(site)g Fo(i)f Fw(is)h(not)g(empt)o(y)l(,)f(then)g(one)g(\(and)h(only)h(one\))-59 480 y(particle)j(c)o(hosen)f(randomly)h(at)f(this)h(site)g(instan)o (tly)g(mo)o(v)o(es)e(to)i(an)o(y)f(site)g Fo(j)j Fw(with)e(probabilit)o (y)h Fo(p)1845 487 y Fm(ij)1876 480 y Fw(.)-59 538 y(T)l(ransitions)20 b(are)e(made)f(indep)q(enden)o(tly)h(for)g(an)o(y)f Fo(i)f Fn(2)g Fo(J)5 b Fw(.)25 b(Probabilities)20 b(of)e(jumps)g(from)f Fo(i)g Fw(to)h Fo(j)j Fw(form)-59 596 y(together)d(single)i(particle)e (la)o(w)h(matrix)f Fh(P)c Fw(=)9 b(\()p Fo(p)842 603 y Fm(ij)872 596 y Fw(\))891 612 y Fm(i;j)r Fv(2)p Fm(J)986 596 y Fo(;)24 b Fn(8)8 b Fo(i)16 b Fn(2)h Fo(J)1193 563 y Fl(P)1183 634 y Fm(j)r Fv(2)p Fm(J)1254 596 y Fo(p)1278 603 y Fm(ij)1325 596 y Fw(=)g(1)p Fo(:)h Fw(One)g(can)g(describ)q(e)g (the)-59 686 y(state)f(of)f(the)g(system)g(b)o(y)g(pro)q(cess)h Fo(\021)r Fw(\()p Fo(t)p Fw(\))d(=)g Fn(f)o Fo(\021)796 693 y Fm(i)811 686 y Fw(\()p Fo(t)p Fw(\))p Fn(g)891 701 y Fm(i)p Fv(2)p Fm(J)959 686 y Fo(;)25 b(t)13 b Fn(\025)h Fw(0)p Fo(;)i Fw(where)g Fo(\021)1301 693 y Fm(i)1316 686 y Fw(\()p Fo(t)p Fw(\))f(is)j(a)e(n)o(um)o(b)q(er)g(of)h(particles) -59 744 y(in)g(cell)g Fo(i)e Fw(at)i(time)f Fo(t)p Fw(.)12 803 y(Let)j Fh(Z)136 810 y Fg(+)184 803 y Fw(=)f Fh(N)11 b Fn([)g(f)p Fh(0)p Fn(g)p Fo(;)p 468 763 35 2 v 37 w Fh(Z)503 810 y Fg(+)551 803 y Fw(=)18 b Fh(Z)641 810 y Fg(+)671 803 y Fn([f1g)p Fo(:)g Fw(Then)h(the)g(state)g(space)g(of)h (our)f(system)f Fh(W)d Fw(=)p 1731 763 V Fh(Z)1765 773 y Fm(J)1765 815 y Fg(+)1814 803 y Fw(is)k(a)-59 861 y(compact)g (metrizable)g(space.)31 b(Let)19 b Fn(B)h Fw(b)q(e)g(a)f Fo(\033)i Fw({)f(algebra)g(generated)f(b)o(y)g(op)q(en)h(sets)f(in)h (the)f(pro)q(duct)-59 919 y(top)q(ology)l(.)k(Let)14 b Fo(C)t Fw(\()p Fh(W)q Fw(\))f(b)q(e)h(the)f(Banac)o(h)h(space)g(of)g (all)i(real{v)m(alued)f(con)o(tin)o(uous)g(functions)g(on)f Fh(W)h Fw(with)-59 977 y(the)h(uniform)h(norm.)12 1035 y(Let)f Fn(8)f Fo(f)k Fn(2)14 b Fo(C)t Fw(\()p Fh(W)q Fw(\))f Fn(8)i Fo(i)e Fn(2)h Fo(J)21 b Fw(the)16 b("measure)h(of)f(dep) q(endence)g(on)g(co)q(ordinate)i Fo(i)p Fw(")e(b)q(e)314 1130 y(\001)355 1137 y Fm(f)377 1130 y Fw(\()p Fo(i)p Fw(\))d(=)h(sup)9 b Fn(fj)k Fo(f)5 b Fw(\()p Fo(\021)r Fw(\))12 b Fn(\000)e Fo(f)5 b Fw(\()p Fo(\020)t Fw(\))15 b Fn(j)p Fw(:)e Fo(\021)r(;)8 b(\020)17 b Fn(2)e Fh(W)q Fo(;)i(\021)1181 1137 y Fm(j)1214 1130 y Fw(=)c Fo(\020)1286 1137 y Fm(j)1321 1130 y Fn(8)i Fo(i)f Fn(6)p Fw(=)f Fo(j)s Fn(g)c Fo(:)-59 1260 y Fw(In)o(tro)q(duce)16 b Fo(D)q Fw(\()p Fh(W)q Fw(\))e(=)366 1187 y Fl(\()399 1260 y Fo(f)19 b Fn(2)c Fo(C)t Fw(\()p Fh(W)q Fw(\))d(:)c Fn(jjj)14 b Fo(f)19 b Fn(jjj)801 1231 y Fm(def)809 1260 y Fw(=)876 1227 y Fl(P)869 1298 y Fm(i)p Fv(2)p Fm(J)936 1260 y Fw(\001)977 1267 y Fm(f)999 1260 y Fw(\()p Fo(i)p Fw(\))13 b Fo(<)h Fn(1)1169 1187 y Fl(\))1211 1260 y Fo(:)h(D)q Fw(\()p Fh(W)q Fw(\))i(is)g(dense)f(in)h Fo(C)t Fw(\()p Fh(W)q Fw(\))-59 1352 y(\()p Fo(see)e Fw([6]\))p Fo(:)h Fw(De\014ne)g(the)g(follo)o(wing)j(op)q(erator)e(on)g Fo(D)q Fw(\()p Fh(W)q Fw(\))d(:)210 1468 y Fo(A)8 b(f)d Fw(\()p Fo(\021)r Fw(\))13 b(=)413 1426 y Fl(X)414 1518 y Fm(i)p Fv(2)p Fm(J)483 1426 y Fl(X)481 1518 y Fm(j)r Fv(2)p Fm(J)552 1468 y Fw([)o Fo(I)t Fn(f)p Fo(\021)640 1475 y Fm(i)668 1468 y Fo(>)h Fw(0)p Fn(g)p Fo(\015)794 1475 y Fm(i)808 1468 y Fo(p)832 1475 y Fm(ij)871 1468 y Fw(\()p Fo(f)5 b Fw(\()p 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b(Op)n(er)n(ator)p 261 2059 37 2 v 18 w Fo(A)c Ff(\(closur)n(e)h(of)f Fo(A)p Ff(\))h(is)f(an)h(in\014nitesimal) h(op)n(er)n(ator)d(for)h(pr)n(o)n(c)n(ess)f Fo(\021)r(:)h(D)q Fw(\()p Fh(W)q Fw(\))h Ff(is)g(a)f(c)n(or)n(e)55 2156 y(of)p 112 2117 V 17 w Fo(A)o(:)-59 2256 y Fh(Pro)r(of:)-59 2314 y Fw(The)g(in\014nitesimal)i(prop)q(erties)e(of)g(an)o(y)f (particle)i(system)e(are)h(describ)q(ed)g(b)o(y)f(the)g(collection)i (of)f(tran-)-59 2380 y(sition)26 b(in)o(tensit)o(y)f(measures)f Fo(c)528 2387 y Fm(T)556 2380 y Fw(\()p Fo(\021)r(;)8 b(\030)r Fw(\))24 b(on)p 765 2340 35 2 v 25 w Fh(Z)800 2350 y Fm(T)800 2392 y Fg(+)857 2380 y Fw(:)g(here)g Fo(\021)29 b Fw(=)f(\()p Fo(\021)1171 2387 y Fm(i)1185 2380 y Fw(\))1204 2394 y Fm(i)p Fv(2)p Fm(J)1272 2380 y Fo(;)36 b(T)e Fw(=)28 b Fn(f)p Fo(i)1493 2387 y Fg(1)1512 2380 y Fo(;)8 b(:)g(:)g(:)f(i)1616 2387 y Fm(n)1640 2380 y Fn(g)24 b Fw(is)h(an)o(y)f(\014-)-59 2438 y(nite)d(subset)h(of)f Fo(J)5 b Fw(,)22 b(and)f Fo(c)446 2445 y Fm(T)474 2438 y Fw(\()p Fo(\021)r(;)8 b(\030)r Fw(\))21 b(is)h(the)f(in)o(tensit)o(y) g(of)h(transition)h(from)e(state)h Fo(\021)h Fw(of)e(the)g(system)f(to) -59 2496 y Fo(\021)-33 2478 y Fm(\030)4 2496 y Fw(=)f(\()p Fo(\020)101 2503 y Fm(j)119 2496 y Fo(;)e(j)f Fn(2)e Fo(J)19 b Fw(:)13 b Fo(\020)327 2503 y Fm(j)360 2496 y Fw(=)g Fo(\021)435 2503 y Fm(j)468 2496 y Fn(8)p Fo(j)21 b(=)-30 b Fn(2)14 b Fo(T)t(;)30 b(\020)677 2503 y Fm(i)689 2509 y Fe(k)724 2496 y Fw(=)14 b Fo(\030)797 2503 y Fm(k)832 2496 y Fn(8)p Fo(k)h Fw(=)f(1)8 b Fo(:)g(:)g(:)g(n)p Fw(\))19 b(in)o(v)o(olving)h(only)g Fo(T)25 b Fw(=)19 b Fn(f)p Fo(i)1592 2503 y Fg(1)1611 2496 y Fo(;)8 b(:)g(:)g(:)g(i)1716 2503 y Fm(n)1739 2496 y Fn(g)19 b Fw(co)q(or-)-59 2554 y(dinates.)j(W)l(e)16 b(ha)o(v)o(e)g(for)14 2636 y Fn(j)p Fo(T)7 b Fn(j)13 b Fo(>)g Fw(2)p Fo(;)22 b Fn(j)p Fo(T)7 b Fn(j)13 b Fw(=)h(1)g(:)101 b Fo(c)505 2643 y Fm(T)532 2636 y Fw(\()p Fo(\021)r(;)8 b(\030)r Fw(\))14 b(=)g(0)14 2753 y Fn(j)p Fo(T)7 b Fn(j)13 b Fw(=)g(2)p Fo(;)22 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Fo(:)1790 2724 y Fw(\(2.2\))904 2936 y Fi(2)p eop %%Page: 3 3 3 2 bop 12 144 a Fw(Construct)16 b Fo(A)8 b(f)d Fw(\()p Fo(\021)r Fw(\))14 b(=)443 111 y Fl(P)452 182 y Fm(T)506 109 y Fl(R)p 491 172 25 2 v 491 200 a Fd(Z)516 179 y Fe(T)516 208 y Fc(+)554 144 y Fo(c)575 151 y Fm(T)602 144 y Fw(\()p Fo(\021)r(;)8 b(d\020)t Fw(\))746 96 y Fl(\020)771 144 y Fo(f)d Fw(\()p Fo(\021)845 126 y Fm(\020)865 144 y Fw(\))11 b Fn(\000)g Fo(f)5 b Fw(\()p Fo(\021)r Fw(\))1038 96 y Fl(\021)1063 144 y Fw(.)21 b(Let)-59 257 y Fo(c)-38 264 y Fm(T)-11 257 y Fw(\()p Fo(i)p Fw(\))c(=)g(sup)q Fn(fk)o Fo(c)260 264 y Fm(T)288 257 y Fw(\()p Fo(\021)331 264 y Fg(1)351 257 y Fo(;)8 b(d\020)t Fw(\))j Fn(\000)g Fo(c)524 264 y Fm(T)551 257 y Fw(\()p Fo(\021)594 264 y Fg(2)614 257 y Fo(;)d(d\020)t Fw(\))p Fn(k)730 272 y Fm(T)766 257 y Fn(j)g Fo(\021)812 264 y Fg(1)832 257 y Fo(;)g(\021)878 264 y Fg(2)915 257 y Fw(:)34 b Fo(\021)987 264 y Fg(1)1006 257 y Fw(\()p Fo(j)s Fw(\))17 b(=)g Fo(\021)1163 264 y Fg(2)1183 257 y Fw(\()p Fo(j)s Fw(\))g Fn(8)p Fo(j)i Fn(6)p Fw(=)e Fo(i)p Fn(g)h Fw(where)g Fn(k\001k)1650 272 y Fm(T)1696 257 y Fw(is)h(a)g(total)-59 328 y(v)m(ariation)h(norm)d (of)h(a)g(measure)f(on)p 634 289 35 2 v 18 w Fh(Z)668 299 y Fm(T)668 340 y Fg(+)697 328 y Fo(:)g Fw(Let)h(us)g(de\014ne)f Fo(\027)s Fw(\()p Fo(i;)8 b(j)s Fw(\))15 b(=)1228 295 y Fl(P)1219 366 y Fm(T)5 b Fv(3)p Fm(i)1288 328 y Fo(c)1309 335 y Fm(T)1337 328 y Fw(\()p Fo(j)s Fw(\))p Fo(;)17 b(i)e Fn(6)p Fw(=)h Fo(j;)h Fw(and)h Fo(\027)s Fw(\()p Fo(i;)8 b(i)p Fw(\))15 b(=)h(0)p Fo(:)-59 414 y Fw(In)g(order)g(to)h (apply)g(existence)f(theorem)f(3.9)i([6])f(w)o(e)f(need)h(to)h(v)o (erify)e(the)h(follo)o(wing)j(conditions:)616 528 y(sup)624 567 y Fm(i)p Fv(2)p Fm(J)698 486 y Fl(X)698 578 y Fm(T)5 b Fv(3)p Fm(i)777 528 y Fw(sup)767 582 y Fm(\021)q Fv(2)p 810 554 25 2 v Fd(Z)834 561 y Fe(T)834 591 y Fc(+)868 528 y Fo(c)889 535 y Fm(T)925 480 y Fl(\020)950 528 y Fo(\021)r(;)p 998 489 35 2 v 8 w Fh(Z)1031 498 y Fm(T)1031 540 y Fg(+)1061 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Fw(of)d(all)g(probabilit)o(y)i(measures)d(on)h (\()p Fh(W)q Fo(;)7 b Fn(B)r Fw(\))16 b(\(see)f([6],)-59 2440 y(prop)q(osition)k(1.8\).)j(Then)446 2427 y(~)441 2440 y Fh(L)14 b Fn(\022)f(M)j Fw(b)o(y)g(de\014nition)i(of)962 2427 y(~)957 2440 y Fh(L)p Fw(.)12 2614 y(Let)791 2631 y Fl(X)792 2723 y Fm(i)p Fv(2)p Fm(J)864 2639 y Fo(\031)892 2646 y Fm(i)p 864 2661 42 2 v 865 2707 a Fo(\015)890 2714 y Fm(i)925 2673 y Fo(<)13 b Fn(1)p Fo(:)750 b Fw(\(3.2\))904 2936 y Fi(4)p eop %%Page: 5 5 5 4 bop -59 140 a Fh(Prop)r(osition)20 b(3.1)i Ff(Supp)n(ose)c(that)f (c)n(onje)n(ctur)n(e)h(A)g(and)f(\(3.2\))g(hold.)23 b(Then)18 b Fn(8)e Fo(\032)e Fn(2)g Fw([0;)8 b Fo(\032)1591 147 y Fg(max)1659 140 y Fw(])651 277 y Fo(L)684 284 y Fm(\032)713 204 y Fl( )746 277 y Fo(\021)15 b Fw(:)821 236 y Fl(X)822 328 y Fm(i)p Fv(2)p Fm(J)889 277 y Fo(\021)913 284 y Fm(i)941 277 y Fo(<)f Fn(1)1043 204 y Fl(!)1090 277 y Fw(=)f(1)p Fo(:)-59 425 y Fh(Theorem)k(3.2)i(\(Clustering\))k Ff(L)n(et)17 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b(c)o(hosen)h(so)-59 1586 y(that)h(matrices)g Fo(P)281 1568 y Fg(\()p Fm(n)p Fg(\))347 1586 y Fw(=)e(\()p Fo(p)443 1593 y Fm(ij)474 1586 y Fw(\))493 1593 y Fm(i;j)r Fv(2)p Fm(J)574 1597 y Fe(n)613 1586 y Fw(are)i(irreducible.)24 b(F)l(or)17 b(giv)o(en)g Fn(f)p Fo(\015)1223 1593 y Fm(i)1237 1586 y Fn(g)1262 1593 y Fm(i)p Fv(2)p Fm(J)1322 1586 y Fw(,)g Fo(P)24 b Fw(and)17 b Fo(\030)h Fn(2)d Fh(Z)1624 1568 y Fm(J)1624 1598 y Fg(+)1670 1586 y Fw(let)i Fo(\021)r Fw(\()p Fo(t)p Fw(\))g(b)q(e)-59 1649 y(the)f(ICJN{pro)q(cess)h (starting)g(from)g Fo(\021)r Fw(\(0\))d(=)f Fo(\030)r Fw(,)k(and)g Fo(\021)959 1631 y Fg(\()p Fm(n)p Fg(\))1010 1649 y Fw(\()p Fo(t)p Fw(\))e(b)q(e)i(the)f(one)g(that)h(starts)g(from) 656 1785 y Fo(\021)682 1765 y Fg(\()p Fm(n)p Fg(\))733 1785 y Fw(\(0\))d(=)861 1712 y Fl(\()915 1756 y Fo(\030)936 1763 y Fm(j)955 1756 y Fo(;)22 b(j)16 b Fn(2)e Fo(J)1101 1763 y Fm(n)1125 1756 y Fo(;)915 1814 y Fn(1)p Fo(;)22 b(j)g(=)-29 b Fn(2)14 b Fo(J)1112 1821 y Fm(n)1135 1814 y Fo(:)-59 1935 y Fw(It)j(follo)o(ws)i(from)e(lemma)g(2.1)h(that)g Fo(\021)r Fw(\()p Fo(t)p Fw(\))c Fn(\036)i Fo(\021)794 1917 y Fg(\()p Fm(n)p Fg(\))845 1935 y Fw(\()p Fo(t)p Fw(\).)23 b(One)17 b(can)h(see)f(that)g(the)g(restriction)i Fo(\021)1665 1910 y Fg(\()p Fm(n)p Fg(\))1663 1947 y Fm(J)1682 1951 y Fe(n)1715 1935 y Fw(\()p Fo(t)p Fw(\))e(is)h(an)-59 2006 y(op)q(ened)f(Jac)o(kson)f(net)o(w)o(ork)g(on)g(the)g(set)h Fo(J)731 2013 y Fm(n)770 2006 y Fw(with)g(input)g(in)o(tensities)h (\001)1282 1988 y Fg(\()p Fm(n)p Fg(\))1346 2006 y Fw(=)c(\(\001)1458 1981 y Fg(\()p Fm(n)p Fg(\))1458 2018 y Fm(j)1509 2006 y Fw(\))1528 2013 y Fm(j)r Fv(2)p Fm(J)1587 2017 y Fe(n)1610 2006 y Fw(,)-59 2079 y(\001)-18 2053 y Fg(\()p Fm(n)p Fg(\))-18 2090 y Fm(j)50 2079 y Fw(=)141 2045 y Fl(P)105 2119 y Fm(i)p Fv(2)p Fm(J)s Fv(n)p Fm(J)200 2123 y Fe(n)230 2079 y Fo(\015)255 2086 y Fm(i)269 2079 y Fo(p)293 2086 y Fm(ij)324 2079 y Fw(.)27 b(Let)18 b Fo(\032)479 2060 y Fg(\()p Fm(n)p Fg(\))548 2079 y 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Fm(J)187 826 y Fe(n)168 853 y Fg(+)227 844 y Fw(w)o(eakly)i(con)o(v)o(erge)g(to)g(the)g(in)o(v)m (arian)o(t)i(measure)512 950 y Fo(\031)540 957 y Fm(u)560 961 y Fe(n)584 950 y Fw(\()p Fo(z)r Fw(\))13 b(=)726 909 y Fl(Y)712 1003 y Fm(j)r Fv(2)p Fm(J)774 994 y Fc(0)771 1012 y Fe(n)792 950 y Fw(\(1)f Fn(\000)f Fo(u)925 957 y Fm(n;j)974 950 y Fw(\))p Fo(u)1021 923 y Fm(z)1037 928 y Fe(j)1021 962 y Fm(n;j)1092 909 y Fl(Y)1078 1003 y Fm(j)r Fv(2)p Fm(J)1140 994 y Fc(1)1137 1012 y Fe(n)1167 950 y Fo(\016)1189 957 y Fv(1)1226 950 y Fw(\()p Fo(z)1268 957 y Fm(j)1286 950 y Fw(\))p Fo(;)-59 1103 y Fw(where)j Fo(\016)102 1110 y Fv(1)139 1103 y Fw(\()p Fn(\001)p Fw(\))h(is)h(a)f(Dirac)h(measure)e(concen)o(trated)g(in)i Fn(1)p Fw(.)21 b(The)14 b(inequalit)o(y)i Fo(u)1391 1110 y Fm(n)1428 1103 y Fn(\024)e Fo(\032)1506 1085 y Fg(\()p Fm(n)p Fg(\))1572 1103 y Fw(yields)i Fo(u)1736 1110 y Fm(n;k)1802 1103 y Fn(!)d Fw(0)-59 1161 y(for)j(an)o(y)h Fo(k)e Fn(2)g Fo(J)223 1168 y Fm(n)244 1173 y Fc(0)263 1161 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2659 y Fw(\026)1004 2672 y Fo(P)1035 2679 y Fm(j)1053 2672 y Fw(\()p Fo(J)5 b Fw(\))14 b(=)g(1)j Ff(for)g(al)r(l)i Fo(j)e Fn(2)d Fo(J)22 b Ff(and)17 b Fw(sup)1611 2711 y Fm(j)r Fv(2)p Fm(J)1687 2672 y Fo(\015)1712 2679 y Fm(j)1745 2672 y Fo(<)c Fn(1)p Ff(.)12 2811 y(Then)18 b(for)e(any)i Fo(\021)r Fw(\(0\))c Fn(2)g Fh(Z)493 2793 y Fm(J)493 2823 y Fg(+)540 2811 y Ff(the)k(pr)n(o)n(c)n(ess)e Fo(\021)r Fw(\()p Fo(t)p Fw(\))d Fn(!)h Fw(0)k Ff(we)n(akly)g(as)f Fo(t)d Fn(!)f(1)p Ff(.)904 2936 y Fi(6)p eop %%Page: 7 7 7 6 bop -59 140 a Fh(Pro)r(of:)-59 198 y Fw(Let)18 b(us)h(v)o(erify)e (that)i(the)f(conditions)j(of)d(prop)q(osition)j(3.2)e(hold)h(in)e(an)o (y)h(of)f(the)g(cases)h(1\))f({)h(4\).)28 b(Then)-59 256 y(immediately)17 b(applying)h(the)e(ab)q(o)o(v)o(e)g(prop)q (osition)j(one)e(can)f(come)g(to)g(the)g(desired)h(conclusion.)12 315 y(1\))d(Because)g(of)g(the)g(transience)h(of)g Fo(P)21 b Fw(the)14 b(sum)g Fo(I)d Fw(+)c Fo(P)14 b Fw(+)7 b Fo(P)1113 296 y Fg(2)1140 315 y Fw(+)g 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1685 y Fe(k)q Fc(+1)1030 1667 y Fw(\))p Fo(p)1073 1647 y Fm(T)1073 1679 y(j)1087 1685 y Fe(l)p Fk(\000)p Fc(1)1139 1679 y Fm(j)1153 1685 y Fe(l)1168 1667 y Fo(\015)1193 1674 y Fm(j)1207 1680 y Fe(l)1221 1667 y Fo(;)-59 1804 y Fw(where)d Fo(p)106 1786 y Fm(T)106 1816 y(ij)153 1804 y Fw(are)g(the)g(elemen)o(ts)g(of)g (the)g(transp)q(ose)i(of)e Fo(P)7 b Fw(:)22 b Fo(p)1029 1786 y Fm(T)1029 1816 y(ij)1073 1804 y Fw(=)14 b Fo(p)1149 1811 y Fm(j)r(i)1180 1804 y Fw(.)21 b(Eviden)o(tly)l(,)478 1935 y Fo(\032)503 1910 y Fg(\()p Fm(n)p Fg(\))503 1947 y Fm(j)568 1935 y Fn(\024)16 b Fw(\026)-26 b Fo(\015)646 1942 y Fm(J)665 1946 y Fe(n)701 1894 y Fl(X)697 1986 y Fv(L)721 1991 y Fe(n;j)766 1935 y Fw(\()789 1881 y Fm(l)p Fv(\000)p Fg(2)791 1894 y Fl(Y)785 1986 y Fm(k)q Fg(=0)858 1935 y Fo(p)882 1915 y Fm(T)882 1948 y(j)896 1954 y Fe(k)915 1948 y Fm(j)929 1954 y Fe(k)q Fc(+1)989 1935 y Fw(\))p Fo(p)1032 1915 y Fm(T)1032 1948 y(j)1046 1954 y Fe(l)p Fk(\000)p Fc(1)1098 1948 y Fm(j)1112 1954 y Fe(l)1141 1935 y Fw(=)15 b(\026)-26 b Fo(\015)1217 1942 y Fm(J)1236 1946 y Fe(n)1260 1935 y Fo(x)1288 1910 y Fg(\()p Fm(n)p Fg(\))1288 1947 y Fm(j)1339 1935 y Fo(;)-59 2082 y Fw(where)25 b(\026)-25 b Fo(\015)115 2089 y Fm(J)134 2093 y Fe(n)185 2082 y Fw(=)50 b(sup)250 2123 y Fm(j)r Fv(2)p Fm(J)s Fv(n)p Fm(J)349 2127 y Fe(n)378 2082 y Fo(\015)403 2089 y Fm(j)422 2082 y Fw(.)44 b(One)24 b(can)g(in)o (terpret)g Fo(x)926 2056 y Fg(\()p Fm(n)p Fg(\))926 2093 y Fm(j)1000 2082 y Fw(as)h(the)f(probabilit)o(y)i(of)e(the)g(ev)o(en)o (t)f(that)h(the)-59 2181 y(terminating)c(Mark)o(o)o(v)e(c)o(hain)h Fo(Y)30 b Fw(with)20 b(state)f(space)g Fo(J)k Fw(and)c(transition)i (matrix)e Fo(P)1518 2162 y Fm(T)1565 2181 y Fw(starting)h(from)f Fo(j)-59 2245 y Fw(ev)o(er)e(en)o(ters)h(the)h(set)f Fo(J)g Fn(n)12 b Fo(J)465 2252 y Fm(n)489 2245 y Fw(.)28 b(In)19 b(view)g(of)g(that)g Fo(x)904 2219 y Fg(\()p Fm(n)p Fg(\))904 2256 y Fm(j)973 2245 y Fn(\024)e Fw(1.)29 b(Hence,)18 b Fo(\032)1283 2219 y Fg(\()p Fm(n)p Fg(\))1283 2256 y Fm(j)1352 2245 y Fn(\024)i Fw(\026)-26 b Fo(\015)1434 2252 y Fm(J)1453 2256 y Fe(n)1477 2245 y Fw(.)28 b(Then)21 b(\026)-26 b Fo(\015)1674 2252 y Fm(J)1693 2256 y Fe(n)1735 2245 y Fn(!)18 b Fw(0)h(as)-59 2317 y Fo(n)14 b Fn(!)g(1)i Fw(b)q(ecause)g(of)h(the)f(con)o(v)o(ergence)e(of)768 2284 y Fl(P)759 2355 y Fm(j)r Fv(2)p Fm(J)829 2317 y Fo(\015)854 2324 y Fm(j)873 2317 y Fw(,)i(that)g(yields)h Fo(\032)1170 2292 y Fg(\()p Fm(n)p Fg(\))1170 2330 y Fm(k)1235 2317 y Fn(!)d Fw(0)j(as)f Fo(n)e Fn(!)g(1)p Fw(.)12 2407 y(3\))g(The)g(ab)q(o)o(v)o(e)g(reasoning)i(holds)g(also)g (for)e(this)h(case,)f(as)h(the)f(requiremen)o(t)h(\026)-26 b Fo(\015)1454 2414 y Fm(J)1473 2418 y Fe(n)1511 2407 y Fn(!)13 b Fw(0)i(is)g(stated)f(no)o(w)-59 2465 y(in)j(the)f (assumptions)i(of)f(3\).)12 2523 y(4\))f(One)g(can)h(adopt)g(the)f(pro) q(of)h(in)g(2\))g(for)f(the)g(case)g Fo(\015)h Fw(=)d Fn(1)p Fw(.)21 b(Namely)l(,)15 b(it)i(is)g(clear)g(that)-59 2590 y Fo(x)-31 2565 y Fg(\()p Fm(n)p Fg(\))-31 2602 y Fm(j)36 2590 y Fn(!)f Fw(1)d Fn(\000)201 2578 y Fw(\026)190 2590 y Fo(P)221 2597 y Fm(j)240 2590 y Fw(\()p Fo(J)5 b Fw(\))17 b(as)h Fo(n)f Fn(!)f(1)p Fw(.)26 b(Then)18 b(it)g(is)h(su\016cien)o(t)f(to)g(require)1264 2578 y(\026)1253 2590 y Fo(P)1284 2597 y Fm(j)1303 2590 y Fw(\()p Fo(J)5 b Fw(\))16 b(=)g(1,)j(sup)1505 2630 y Fm(j)r Fv(2)p Fm(J)1581 2590 y Fo(\015)1606 2597 y Fm(j)1641 2590 y Fo(<)e Fn(1)h Fw(to)g(get)-59 2695 y Fo(\032)-34 2669 y Fg(\()p Fm(n)p Fg(\))-34 2706 y Fm(j)31 2695 y Fn(!)c Fw(0.)904 2936 y Fi(7)p eop %%Page: 8 8 8 7 bop -59 140 a Fh(De\014nition)19 b(3.1)k Ff(The)g Fo(i)p Ff(-th)h(c)n(o)n(or)n(dinate)e Fo(\021)764 147 y Fm(i)778 140 y Fw(\()p Fn(\001)p Fw(\))h Ff(of)g(the)h(pr)n(o)n(c)n (ess)d Fo(\021)k Ff(\(the)f Fo(i)p Ff(-th)f(c)n(el)r(l)i(of)e(the)h (system\))f(is)-59 198 y Fw(sto)q(c)o(hastically)c(b)q(ounded)f Ff(if)34 b Fw(lim)491 223 y Fm(m)p Fv(!1)611 198 y Fw(sup)601 238 y Fm(t)p Fv(2)p Fd(R)668 242 y Fc(+)702 198 y Fo(P)7 b Fn(f)p Fo(\021)789 205 y Fm(i)803 198 y Fw(\()p Fo(t)p Fw(\))13 b Fo(>)h(m)p Fn(g)g Fw(=)f(0)p Fo(:)-59 332 y Fh(Theorem)k(3.4)i(\(Sto)r(c)n(hastical)h(b)r(oundedness\))i Ff(If)c Fn(9)p Fo(i)1054 339 y Fg(0)1088 332 y Fn(2)d Fo(J)5 b Ff(:)23 b Fo(\015)1231 339 y Fm(i)1243 344 y Fc(0)1278 332 y Fo(>)1340 299 y Fl(P)1331 370 y Fm(j)r Fv(2)p Fm(J)1401 332 y Fo(\015)1426 339 y Fm(j)1445 332 y Fo(p)1469 339 y Fm(j)r(i)1497 344 y Fc(0)1535 332 y Ff(and)18 b Fo(\021)1654 339 y Fm(i)1666 344 y Fc(0)1686 332 y Fw(\(0\))d Fn(2)g Fh(Z)1845 339 y Fg(+)1875 332 y Ff(,)-59 422 y(then)j Fo(\021)73 429 y Fm(i)85 434 y Fc(0)105 422 y Fw(\()p Fo(t)p Fw(\))f Ff(is)g(sto)n(chastic)n(al)r (ly)h(b)n(ounde)n(d,)g(and)565 576 y Fw(lim)8 b(sup)598 613 y Fm(t)p Fv(!1)722 576 y Fo(E)s(\021)785 583 y Fm(i)797 588 y Fc(0)817 576 y Fw(\()p Fo(t)p Fw(\))13 b Fn(\024)1012 475 y Fl(P)958 547 y Fm(j)r Fv(2)p Fm(J)o(;)d(j)r Fv(6)p Fg(=)p Fm(i)1091 552 y Fc(0)1117 508 y Fo(\015)1142 515 y Fm(j)1161 508 y Fo(p)1185 515 y Fm(j)r(i)1213 520 y Fc(0)p 944 564 304 2 v 944 610 a Fo(\015)969 617 y Fm(i)981 622 y Fc(0)1012 610 y Fn(\000)1071 577 y Fl(P)1061 648 y Fm(j)r Fv(2)p Fm(J)1132 610 y Fo(\015)1157 617 y Fm(j)1175 610 y Fo(p)1199 617 y Fm(j)r(i)1227 622 y Fc(0)1252 576 y Fo(:)-59 738 y Fh(Pro)r(of:)-59 796 y Fw(Construct)17 b(the)f(pro)q(cess)h Fo(\021)449 778 y Fv(0)460 796 y Fw(\()p Fo(t)p Fw(\))f(suc)o(h)g(that)653 924 y Fo(\021)679 903 y Fv(0)691 924 y Fw(\(0\))e(=)819 851 y Fl(\()873 894 y Fo(\021)897 901 y Fm(j)915 894 y Fw(\(0\))p Fo(;)22 b(j)17 b Fw(=)d Fo(i)1119 901 y Fg(0)1138 894 y Fo(;)873 952 y Fn(1)p Fo(;)21 b(j)c Fn(6)p Fw(=)d Fo(i)1064 959 y Fg(0)1083 952 y Fo(:)-59 1052 y Fw(Again,)j(using)h(lemma)e(2.1,)g(w) o(e)g(obtain)i Fo(\021)r Fw(\()p Fo(t)p Fw(\))13 b Fn(\036)h Fo(\021)872 1034 y Fv(0)884 1052 y Fw(\()p Fo(t)p Fw(\).)21 b(Note)16 b(that)h Fo(\021)1225 1034 y Fv(0)1236 1052 y Fw(\(0\))g(is)g(an)g(op)q(ened)g(\014nite)g(Jac)o(kson)-59 1119 y(net)o(w)o(ork)i(with)i(only)f(one)g(no)q(de)h Fo(i)587 1126 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1280 y Fo(p)1601 1287 y Fm(i)1613 1292 y Fc(0)1631 1287 y Fm(i)1643 1292 y Fc(0)1662 1280 y Fw(,)14 b(and)h Fo(\021)1809 1262 y Fv(0)1807 1293 y Fm(i)1819 1298 y Fc(0)1838 1280 y Fw(\()p Fn(\001)p Fw(\))-59 1352 y(is)i(ergo)q(dic)g(i\013)h Fo(\032)243 1326 y Fg(\(1\))243 1363 y Fm(i)255 1368 y Fc(0)304 1352 y Fo(<)c(\015)381 1359 y Fm(i)393 1364 y Fc(0)412 1352 y Fw(,)i(i.)22 b(e.,)15 b(\001)597 1326 y Fg(\(1\))597 1363 y Fm(i)609 1368 y Fc(0)644 1352 y Fo(=)p Fw(\(1)c Fn(\000)g Fo(p)796 1359 y Fm(i)808 1364 y Fc(0)826 1359 y Fm(i)838 1364 y Fc(0)857 1352 y Fw(\))j Fo(<)g(\015)967 1359 y Fm(i)979 1364 y Fc(0)999 1352 y Fw(,)i(or)651 1413 y Fl(X)608 1505 y Fm(j)r Fv(2)p Fm(J)o(;)6 b(j)r Fv(6)p Fg(=)p Fm(i)737 1510 y Fc(0)763 1454 y Fo(\015)788 1461 y Fm(j)807 1454 y Fo(p)831 1461 y Fm(j)r(i)859 1466 y Fc(0)892 1454 y Fo(<)14 b(\015)969 1461 y Fm(i)981 1466 y Fc(0)1001 1454 y Fw(\(1)d Fn(\000)g Fo(p)1129 1461 y Fm(i)1141 1466 y Fc(0)1159 1461 y Fm(i)1171 1466 y Fc(0)1190 1454 y Fw(\))p Fo(:)-59 1594 y Fw(The)16 b(last)i(inequalit)o(y)f(holds)g(due)g(to)f(the)g(assumptions)i(of)f (the)f(theorem.)k(Consequen)o(tly)l(,)c Fo(\021)1686 1576 y Fv(0)1684 1606 y Fm(i)1696 1611 y Fc(0)1716 1594 y Fw(\()p Fn(\001)p Fw(\))g(is)h(er-)-59 1652 y(go)q(dic)g(and)f(sto)q (c)o(hastically)i(b)q(ounded.)k(Therefore,)15 b Fo(\021)941 1659 y Fm(i)953 1664 y Fc(0)972 1652 y Fw(\()p Fn(\001)p Fw(\))h(is)g(sto)q(c)o(hastically)i(b)q(ounded.)k(Let)16 b Fo(\021)1721 1634 y Fv(0)1719 1664 y Fm(i)1731 1669 y Fc(0)1750 1652 y Fw(\()p Fn(1)p Fw(\))e(=)-51 1723 y(lim)-59 1750 y Fm(t)p Fv(!1)33 1723 y Fo(\021)59 1705 y Fv(0)57 1735 y Fm(i)69 1740 y Fc(0)88 1723 y Fw(\()p Fo(t)p Fw(\).)19 b(Then)12 b Fo(\021)326 1705 y Fv(0)324 1735 y Fm(i)336 1740 y Fc(0)356 1723 y Fw(\()p Fn(1)p Fw(\))f(has)h(a)g(geometrical)h(distribution)h Fo(P)1137 1674 y Fl(n)1179 1693 y Fv(\030)1165 1723 y Fo(\021)1189 1730 y Fm(i)1201 1735 y Fc(0)1234 1723 y Fw(\()p Fn(1)p Fw(\))f(=)h Fo(n)1416 1674 y Fl(o)1458 1723 y Fw(=)g(\(1)r Fn(\000)r Fo(\013)p Fw(\))p Fo(\013)1677 1705 y Fm(n)1700 1723 y Fo(;)22 b(n)14 b Fn(2)g Fh(Z)1860 1730 y Fg(+)-59 1796 y Fw(with)j(parameter)f(\(see)g([1]\))718 1913 y Fo(\013)e Fw(=)873 1812 y Fl(P)820 1884 y Fm(j)r Fv(2)p Fm(J)o(;)c(j)r Fv(6)p Fg(=)p Fm(i)953 1889 y Fc(0)979 1845 y Fo(\015)1004 1852 y Fm(j)1022 1845 y Fo(p)1046 1852 y Fm(j)r(i)1074 1857 y Fc(0)p 820 1901 275 2 v 825 1947 a Fw(\(1)h Fn(\000)g Fo(p)953 1954 y Fm(i)965 1959 y Fc(0)983 1954 y Fm(i)995 1959 y Fc(0)1014 1947 y Fw(\))p Fo(\015)1058 1954 y Fm(i)1070 1959 y Fc(0)1099 1913 y Fo(:)-59 2029 y Fw(It)16 b(follo)o(ws)i(from)e Fo(\021)r Fw(\()p Fo(t)p Fw(\))d Fn(\036)h Fo(\021)443 2011 y Fg(\()p Fm(n)p Fg(\))494 2029 y Fw(\()p Fo(t)p Fw(\))h(that)i Fo(E)s(\021)734 2036 y Fm(i)746 2041 y Fc(0)766 2029 y Fw(\()p Fo(t)p Fw(\))c Fn(\024)h Fo(E)s(\021)953 2011 y Fv(0)951 2041 y Fm(i)963 2046 y Fc(0)982 2029 y Fw(\()p Fo(t)p Fw(\))i(for)g(all)i Fo(t)c Fn(\025)f Fw(0)k(and)534 2142 y(lim)8 b(sup)567 2179 y Fm(t)p Fv(!1)692 2142 y Fo(E)s(\021)755 2149 y Fm(i)767 2154 y Fc(0)787 2142 y Fw(\()p Fo(t)p Fw(\))13 b Fn(\024)g Fo(E)s(\021)973 2121 y Fv(0)971 2154 y Fm(i)983 2159 y Fc(0)1003 2142 y Fw(\()p Fn(1)p Fw(\))h(=)1204 2108 y Fo(\013)p 1161 2130 117 2 v 1161 2176 a Fw(1)e Fn(\000)f Fo(\013)1283 2142 y(:)-59 2258 y Fw(Substituting)18 b Fo(\013)p Fw(,)e(w)o(e)g (obtain)i(the)e(last)h(statemen)o(t)f(of)h(the)f(theorem.)-59 2406 y Fp(4)67 b(Case)21 b Fb(\015)f(<)c Fa(1)-59 2514 y Fw(In)c(this)i(case)f(one)g(can)g(pro)o(v)o(e)f(that)h(the)g (transitions)i(of)e(particles)h(o)q(ccur)f(a.)20 b(s.)h(only)13 b(in)h(discrete)e(random)-59 2572 y(momen)o(ts)j(of)h(time)g Fo(t)332 2579 y Fm(n)355 2572 y Fo(;)24 b(t)411 2579 y Fm(n)445 2572 y Fn(\000)10 b Fo(t)512 2579 y Fm(n)p Fv(\000)p Fg(1)594 2572 y Fw(=)j Fo(\034)666 2579 y Fm(n)704 2572 y Fn(\030)j Fw(i.)21 b(i.)h(exp)q(onen)o(tially)17 b(d.)22 b(r.)f(v.)g(with)16 b(parameter)g Fo(\015)h Fw(=)1778 2539 y Fl(P)1771 2610 y Fm(i)p Fv(2)p Fm(J)1837 2572 y Fo(\015)1862 2579 y Fm(i)1876 2572 y Fo(;)-59 2655 y(n)f Fw(|)g(n)o(um)o(b)q(er)g(of)g(transition)j(since)d Fo(t)e Fw(=)g(0)p Fo(:)i Fw(Then)81 2771 y Fo(P)g Fn(f)o Fo(n)p Fw({th)h(transition)i(is)e(made)f(from)g(cell)h Fo(i)c Fn(j)h Fo(\021)954 2778 y Fm(i)968 2771 y Fw(\()p Fo(t)1005 2778 y Fm(n)1039 2771 y Fn(\000)d Fw(0\))j Fo(>)g Fw(0)p Fn(g)g Fw(=)1318 2737 y Fo(\015)1343 2744 y Fm(i)p 1318 2759 40 2 v 1324 2805 a Fo(\015)1376 2742 y Fm(def)1385 2771 y Fw(=)22 b Fo(\014)1473 2778 y Fm(i)1487 2771 y Fo(;)1536 2730 y Fl(X)1537 2821 y Fm(i)p Fv(2)p Fm(J)1604 2771 y Fo(\014)1632 2778 y Fm(i)1660 2771 y Fw(=)14 b(1)p Fo(:)904 2936 y Fi(8)p eop %%Page: 9 9 9 8 bop -59 140 a Fw(In)o(tro)q(duce)14 b(em)o(b)q(edded)g(Mark)o(o)o (v)g(c)o(hain)h Fo(\021)r Fw(\()p Fo(n)p Fw(\))f(=)g Fn(f)p Fo(\021)900 147 y Fm(i)914 140 y Fw(\()p Fo(t)951 147 y Fm(n)974 140 y Fw(\))p Fn(g)1018 155 y Fm(i)p Fv(2)p Fm(J)1101 140 y Fn(8)p Fo(n)p Fn(2)f Fh(N)o Fw(.)21 b(Later)15 b(on)h(w)o(e)e(shall)i(write)g Fo(\021)1809 147 y Fm(i)1823 140 y Fw(\()p Fo(n)p Fw(\))-59 198 y(instead)j(of)f Fo(\021)192 205 y Fm(i)207 198 y Fw(\()p Fo(t)244 205 y Fm(n)267 198 y Fw(\).)26 b(Supp)q(ose)20 b Fo(\021)544 205 y Fm(i)558 198 y Fw(\(0\))d(=)g Fo(x)720 205 y Fm(i)751 198 y Fn(8)g Fo(i;)854 165 y Fl(P)847 236 y Fm(i)p Fv(2)p Fm(J)914 198 y Fo(x)942 205 y Fm(i)972 198 y Fw(=)g Fn(1)p Fo(:)h Fw(Let)g(us)h(in)o(tro)q(duce)f(the)g(non{decreasing)-59 284 y(family)f(of)g Fo(\033)r Fn(\000)p Fw(algebras)g Fn(fF)465 291 y Fm(n)488 284 y Fn(g)513 291 y Fm(n)p Fv(2)p Fd(N)592 284 y Fw(,)325 378 y Fn(F)361 385 y Fm(n)398 378 y Fw(=)d Fo(\033)c Fw(\()p Fo(\021)r Fw(\()p Fo(m)p Fw(\))p Fo(;)23 b(m)14 b Fn(\024)f Fo(n)p Fw(\))h(=)g Fo(\033)c Fw(\()p Fo(\021)955 385 y Fm(i)969 378 y Fw(\()p Fo(m)p Fw(\))p Fo(;)23 b(i)14 b Fn(2)g Fo(J)o(;)24 b(m)14 b Fn(\024)f Fo(n)p Fw(\))41 b Fn(8)p Fo(n)8 b(:)-59 476 y Fh(De\014nition)19 b(4.1)k Ff(Me)n(asur)n(e)18 b Fo(\026)p Fw(\()p Fn(\001)p Fw(\))h Ff(on)g Fo(J)k Ff(\(p)n(ossibly)c Fo(\033)r Ff(-\014nite\))h(is)f(c)n(al)r(le)n(d)h(\(strictly\))f(exc)n (essive)h(for)f(tr)n(an-)-59 534 y(sition)f(matrix)f Fo(P)k Fw(=)14 b(\()p Fo(p)380 541 y Fm(ij)411 534 y Fw(\))430 541 y Fm(i;j)r Fv(2)p Fm(J)533 534 y Ff(i\013)35 b Fo(\026)14 b Fn(\025)f Fo(\026P)22 b Fw(:)f Fn(8)p Fo(i)15 b(\026)915 541 y Fm(i)944 534 y Fn(\025)1005 501 y Fl(P)996 572 y Fm(j)r Fv(2)p Fm(J)1067 534 y Fo(\026)1096 541 y Fm(j)1114 534 y Fo(p)1138 541 y Fm(j)r(i)1186 534 y Ff(\()p Fo(\026)g(>)e(\026P)7 b Ff(,)18 b(r)n(esp)n(e)n(ctively\).) -59 664 y Fh(De\014nition)h(4.2)k Ff(F)l(unction)15 b Fo(f)20 b Ff(on)14 b Fo(J)k Ff(is)c(c)n(al)r(le)n(d)h(harmonious)f(\(r) n(esp.)20 b(exc)n(essive\))c(for)d(tr)n(ansition)h(matrix)-59 722 y Fo(P)21 b Fw(=)14 b(\()p Fo(p)88 729 y Fm(ij)118 722 y Fw(\))137 729 y Fm(i;j)r Fv(2)p Fm(J)241 722 y Ff(i\013)j Fo(P)7 b(f)20 b Fw(=)13 b Fo(f)20 b Fw(:)h Fn(8)p Fo(i)15 b(f)600 729 y Fm(i)628 722 y Fw(=)689 689 y Fl(P)680 760 y Fm(j)r Fv(2)p Fm(J)750 722 y Fo(p)774 729 y Fm(ij)805 722 y Fo(f)829 729 y Fm(j)865 722 y Ff(\()p Fo(P)7 b(f)19 b Fn(\024)14 b Fo(f)5 b Ff(\).)-59 852 y Fh(Lemma)18 b(4.1)k Ff(L)n(et)17 b Fo(\026)p Fw(\()p Fn(\001)p Fw(\))g Ff(and)h Fo(\027)s Fw(\()p Fn(\001)p Fw(\))g Ff(b)n(e)f(\014nite)i(me)n(asur)n(es)e(on)g(the)h(same)g(me)n (asur)n(able)f(sp)n(ac)n(e)g Fw(\(\012)p Fo(;)8 b Fn(L)p Fw(\))q Ff(,)-59 910 y Fo(\026)p Fw(\(\012\))14 b(=)g Fo(\027)s Fw(\(\012\))g(=)g Fo(a)g(<)f Fn(1)p Ff(.)22 b(If)c Fn(8)p Fo(A)12 b Fn(2)i(L)50 b Fo(\027)s Fw(\()p Fo(A)p Fw(\))13 b Fn(\025)h Fo(\026)p Fw(\()p Fo(A)p Fw(\))p Ff(,)j(then)i Fo(\026)p Fw(\()p Fn(\001)p Fw(\))14 b(=)f Fo(\027)s Fw(\()p Fn(\001)p Fw(\))p Ff(.)-59 1008 y Fh(Pro)r(of:)-59 1066 y Fw(Supp)q(ose)k(the)f(con)o(trary)g(holds:)23 b Fn(9)8 b Fo(A)13 b Fn(2)h(L)h Fw(:)21 b Fo(\027)s Fw(\()p Fo(A)p Fw(\))14 b Fo(>)g(\026)p Fw(\()p Fo(A)p Fw(\))p Fo(:)h Fw(But)h(b)o(y)g(condition)i(of)e(the)g(lemma)-59 1124 y Fo(\027)s Fw(\()0 1111 y(\026)-13 1124 y Fo(A)p Fw(\))e Fn(\025)f Fo(\026)p Fw(\()170 1111 y(\026)157 1124 y Fo(A)p Fw(\).)21 b(Summing)16 b(these)f(inequalities)j(up)d(one) h(can)f(obtain)i Fo(a)d(>)g(a)p Fw(.)20 b(W)l(e)15 b(arriv)o(ed)h(at)g (a)f(con)o(tra-)-59 1182 y(diction.)22 b(Th)o(us)17 b(the)f(statemen)o (t)g(of)g(lemma)g(is)h(pro)o(v)o(ed.)-59 1327 y Fh(Lemma)h(4.2)k Ff(If)d(c)n(onje)n(ctur)n(e)h(B)f(holds)h(and)g Fn(f)p Fo(\014)849 1334 y Fm(i)862 1327 y Fn(g)887 1334 y Fm(i)p Fv(2)p Fm(J)965 1327 y Fn(6)p Fw(=)d Fn(f)p Fo(\031)1073 1334 y Fm(i)1087 1327 y Fn(g)1112 1334 y Fm(i)p Fv(2)p Fm(J)1191 1327 y Ff(or)i(if)g(c)n(onje)n(ctur)n(e)h(B)g(do)n(es)f(not)h (hold,)-59 1385 y(then)452 1444 y Fn(9)8 b Fo(i)505 1451 y Fg(0)525 1444 y Fo(;)16 b(j)575 1451 y Fg(0)608 1444 y Fw(:)64 b Fo(\015)711 1451 y Fm(i)723 1456 y Fc(0)756 1444 y Fo(>)809 1402 y Fl(X)808 1494 y Fm(j)r Fv(2)p Fm(J)878 1444 y Fo(\015)903 1451 y Fm(j)922 1444 y Fo(p)946 1451 y Fm(j)r(i)974 1456 y Fc(0)994 1444 y Fo(;)58 b(\015)1091 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Fm(j)r Fv(2)p Fm(J)116 2779 y Fo(\015)141 2786 y Fm(j)160 2779 y Fo(p)184 2786 y Fm(j)r(i)264 2779 y Fn(8)p Fo(i)d Fn(2)h Fo(J)22 b Ff(al)r(l)c(c)n(o)n(or)n(dinates)f Fo(\021)766 2786 y Fm(i)780 2779 y Fw(\()p Fo(t)p Fw(\))g Ff(ar)n(e)g(sto)n(chastic)n(al)r (ly)h(b)n(ounde)n(d.)904 2936 y Fi(9)p eop %%Page: 10 10 10 9 bop -59 140 a Fh(Pro)r(of:)-59 198 y Fw(If)19 b(conjecture)f(C)i (holds,)g(then)g(for)f(some)g Fo(i)g(G)820 205 y Fm(ii)866 198 y Fo(<)g Fn(1)p Fw(.)30 b(The)19 b(assumptions)i(concerning)f(the)f (prop)q(er-)-59 256 y(ties)h(of)g(a)f(Mark)o(o)o(v)g(c)o(hain)h(with)g (the)g(transition)h(matrix)f Fo(P)27 b Fw(guaran)o(tee)19 b(the)h(last)g(inequalit)o(y)h(to)f(hold)-59 315 y Fn(8)p Fo(i)d Fn(2)h Fo(J)5 b Fw(.)29 b(Then)19 b(for)g(an)o(y)g Fn(f)p Fo(\027)481 322 y Fm(i)495 315 y Fn(g)520 322 y Fm(i)p Fv(2)p Fm(J)598 315 y Fw(:)26 b Fo(\027)662 322 y Fm(i)695 315 y Fn(\025)18 b Fw(0)49 b Fn(8)p Fo(i)17 b Fn(2)i Fo(J)k Fw(construct)c(the)g(measure)f Fo(\026)h Fw(=)f Fn(f)p 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Fo(P)966 1096 y Fl(n)994 1144 y Fo(\034)1021 1119 y Fg(\()p Fm(i)p Fg(\))1015 1155 y(1)1062 1144 y Fw(\()p Fo(n)p Fw(\))g(=)g Fo(k)1222 1096 y Fl(o)1258 1144 y Fo(:)f Fw(Then)h(the)g(follo)o(wing)j(recurren) o(t)-59 1225 y(form)o(ulae)f(could)i(b)q(e)e(deduced)g(b)o(y)f(means)i (of)f(the)g(total)i(probabilit)o(y)g(form)o(ula:)417 1366 y Fo(F)456 1345 y Fm(n)449 1378 y(i)478 1366 y Fw(\()p Fo(z)r Fw(\))c(=)607 1266 y Fl(8)607 1304 y(>)607 1316 y(<)607 1391 y(>)607 1403 y(:)665 1307 y Fo(pz)r(F)753 1289 y Fm(n)746 1319 y(i)p Fg(+1)805 1307 y Fw(\()p Fo(z)r Fw(\))d(+)g Fo(q)r(z)r(F)1016 1289 y Fm(n)1009 1319 y(i)p Fv(\000)p Fg(1)1067 1307 y Fw(\()p Fo(z)r Fw(\))p Fo(;)21 b Fw(1)14 b Fo(<)g(i)g(<)f(n)665 1365 y Fw(1)p Fo(;)22 b(i)13 b Fw(=)h(1)665 1423 y(0)p Fo(;)22 b(i)13 b Fw(=)h Fo(n)1401 1366 y(:)375 b Fw(\(4.7\))-59 1548 y(Solving)20 b(this)f(equation)f(one)g(can)h(see)e(that)i Fo(F)822 1530 y Fm(n)815 1560 y(i)844 1548 y Fw(\()p Fo(z)r Fw(\))e(=)g Fo(A)p Fw(\()p Fo(z)r Fw(\))1087 1487 y Fl(\022)1122 1520 y Fg(1)p Fv(\000)1167 1485 y Fn(p)p 1208 1485 133 2 v 1208 1520 a Fg(1)p Fv(\000)p Fg(4)p Fm(pq)q(z)1323 1510 y Fc(2)p 1122 1536 220 2 v 1205 1565 a Fg(2)p Fm(pz)1346 1487 y Fl(\023)1376 1494 y Fm(i)1403 1548 y Fw(+)12 b Fo(B)s Fw(\()p Fo(z)r Fw(\))1564 1487 y Fl(\022)1599 1520 y Fg(1+)1644 1485 y Fn(p)p 1686 1485 133 2 v 35 x Fg(1)p Fv(\000)p Fg(4)p Fm(pq)q(z)1801 1510 y Fc(2)p 1599 1536 220 2 v 1682 1565 a Fg(2)p Fm(pz)1823 1487 y Fl(\023)1854 1494 y Fm(i)1876 1548 y Fo(;)-59 1627 y Fw(where)k Fo(A)p Fw(\()p Fo(z)r Fw(\))f(and)i Fo(B)s Fw(\()p Fo(z)r Fw(\))f(could)h(b)q(e)f(found)h(from)f(b)q(oundary)i (conditions)g(\(4.7\).)k(And)16 b(as)-59 1716 y Fo(P)f Fn(f)p Fo(C)47 1723 y Fm(n)71 1716 y Fn(g)e Fw(=)h Fo(F)200 1698 y Fm(n)193 1728 y(i)223 1716 y Fw(\(1\))j(one)f(can)h(calculate)g (that)f Fo(P)g Fn(f)p Fo(C)896 1723 y Fm(n)919 1716 y Fn(g)e Fw(=)1015 1687 y(\()1039 1667 y Fe(q)p 1039 1675 17 2 v 1039 1696 a(p)1060 1687 y Fw(\))1079 1660 y Fe(n)1100 1683 y Fv(\000)1127 1687 y Fw(\()1152 1667 y Fe(q)p 1151 1675 V 1151 1696 a(p)1172 1687 y Fw(\))1191 1660 y Fe(i)p 1015 1704 190 2 v 1040 1743 a Fw(\()1065 1724 y Fe(q)p 1064 1732 17 2 v 1064 1752 a(p)1085 1743 y Fw(\))1104 1716 y Fe(n)1125 1739 y Fv(\000)1157 1724 y Fe(q)p 1158 1732 V 1158 1752 a(p)1229 1716 y Fn(\000)-8 b(!)1223 1741 y Fm(n)p Fv(!1)1329 1667 y Fl(\020)1359 1694 y Fm(q)p 1359 1704 18 2 v 1359 1733 a(p)1382 1667 y Fl(\021)1406 1679 y Fm(i)p Fv(\000)p Fg(1)1474 1716 y Fo(:)16 b Fw(Then)-48 1780 y Fv(1)-52 1788 y Fl(P)-59 1858 y Fm(i)p Fg(=1)6 1821 y Fo(G)44 1828 y Fm(i;)p Fg(1)100 1821 y Fw(=)157 1802 y Fg(1)p 157 1810 V 158 1838 a Fm(b)199 1780 y Fv(1)195 1788 y Fl(P)188 1858 y Fm(i)p Fg(=1)253 1773 y Fl(\020)283 1800 y Fm(q)p 283 1810 V 283 1838 a(p)306 1773 y Fl(\021)331 1785 y Fm(i)p Fv(\000)p Fg(1)404 1821 y Fw(=)504 1800 y Fm(p)p 460 1810 105 2 v 460 1838 a(b)p Fg(\()p Fm(p)p Fv(\000)p Fm(q)q Fg(\))584 1821 y Fo(<)e Fn(1)p Fo(;)h 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(V.)f(I.)g(Oseledets)i("Mean-\014eld)g(appro)o(ximation)h(for)f(sto)q (c)o(hastic)g(transp)q(ortation)40 2753 y(net)o(w)o(ork)d(and)i (stabilit)o(y)h(of)f(dynamical)h(system")e Ff(Pr)n(eprint)h(N)h(445)g (of)g(the)g(University)h(of)e(Br)n(emen,)40 2811 y(Br)n(emen,)k(June)h (1999)893 2936 y Fi(12)p eop %%Page: 13 13 13 12 bop -35 140 a Fw([4])23 b(V.)16 b(Sc)o(herbak)o(o)o(v)f("Time)j (scales)f(hierarc)o(h)o(y)f(in)i(large)g(closed)f(Jac)o(kson)g(net)o(w) o(orks")g Ff(Pr)n(eprint)h(N)g(4,)40 198 y(F)l(r)n(ench-R)o(ussian)f (A.M.)g(Liapunov)h(Institute)h(of)e(Mosc)n(ow)g(State)i(University,)f (Mosc)n(ow,)f(1997)-35 294 y Fw([5])23 b(E.)c(Sp)q(o)q(darev)h("T)l (ransp)q(ort)h(net)o(w)o(orks)e(with)i(an)e(in\014nite)i(n)o(um)o(b)q (er)d(of)i(no)q(des")g Ff(to)h(app)n(e)n(ar)e(in)h(the)40 352 y(c)n(onfer)n(enc)n(e)e(issue)g(of)f(Zeitschrift)h(f)q(\177)-26 b(ur)17 b(angewandte)i(Mathematik)f(und)g(Me)n(chanik,)g(1999)-35 447 y Fw([6])23 b(T.)16 b(M.)f(Liggett)j("In)o(teracting)e(particle)i 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y(coun)o(table)16 b(Mark)o(o)o(v)g(c)o(hains")h Ff(Cambridge)h(University)g(Pr)n(ess,)f (1995)-59 1867 y Fw([16])23 b(L.)18 b(Afanas'ev)m(a,)g(E.)g(Bulinsk)m (a)o(y)o(a)h("Sto)q(c)o(hastic)h(pro)q(cesses)f(in)g(queueing)f(and)h (in)o(v)o(en)o(tory)f(theory")40 1925 y Ff(Mosc)n(ow,)f(Izd.)g(MGU,)g (1980)g(\(in)g(russian\))-59 2021 y Fw([17])23 b(E.)11 b(Bulinsk)m(a)o(y)o(a)h("On)g(high)h({)f(lev)o(el)g(crossing)h(for)f(a) g(class)g(of)g(discrete)g({)g(time)f(sto)q(c)o(hastic)i(pro)q(cesses") 40 2079 y Ff(F)l(undamentalnaya)18 b(i)g(prikladnaya)f(matematika)h(1,) g(81-107)f(\(1995\))893 2936 y Fi(13)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF