Content-Type: multipart/mixed; boundary="-------------0305010554172" This is a multi-part message in MIME format. ---------------0305010554172 Content-Type: text/plain; name="03-205.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-205.keywords" stochastic processes, non-Markovian processes, Gibbs measures ---------------0305010554172 Content-Type: application/postscript; name="ccc-gibbs.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ccc-gibbs.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ccc-gibbs.dvi %%CreationDate: Thu May 01 12:15:37 2003 %%Pages: 31 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: DVIPS.EXE ccc-gibbs %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.05.01:1215 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}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{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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y(\(Theorem)i(A.4\).) 257 3434 y Fw(2)161 b(Notation)54 b(and)g(preliminary)h(de\014nitions) 257 3653 y Fv(W)-8 b(e)46 b(consider)g(a)f(\014nite)g(alphab)s(et)g Fr(A)g Fv(endo)m(w)m(ed)i(with)e(the)h(discrete)g(top)s(ology)e(and)257 3773 y Fq(\033)t Fv(-algebra,)g(and)f(\012)h(a)e(compact)h(subset)h(of) f Fr(A)2047 3737 y Fp(Z)2095 3773 y Fv(.)74 b(The)44 b(space)g(\012)f(is)g(endo)m(w)m(ed)i(with)257 3893 y(the)k(pro)5 b(jection)48 b Fr(F)58 b Fv(of)48 b(the)g(pro)s(duct)h Fq(\033)t Fv(-algebra)d(asso)s(ciated)i(to)g Fr(A)2885 3857 y Fp(Z)2933 3893 y Fv(.)91 b(The)49 b(space)257 4014 y(\012)d(represen)m(ts)i(admissible)43 b(\\letter)h (con\014gurations",)k(where)e(the)g(admissibilit)m(y)c(is)257 4134 y(de\014ned,)54 b(for)48 b(instance)h(b)m(y)g(some)g(exclusion)f (rule)g(as)h(in)e(Ruelle)h(\(1978\))f(or)h(b)m(y)h(a)257 4255 y(\\grammar")35 b(\(subshift)j(of)f(\014nite)g(t)m(yp)s(e\))h(as)f (in)g(W)-8 b(alters)37 b(\(1975\).)57 b(F)-8 b(or)37 b(eac)m(h)h(\003)d Fr(\032)i Fo(Z)p 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Fg(,)f Fq(f)1066 5009 y Fn(\003)1147 4994 y Fv(:)27 b Fr(F)1273 5009 y Fl(\024)p Fm(m)1390 5020 y Ff(\003)1462 4994 y Fr(\002)c Fv(\012)28 b Fr(\000)-16 b(!)27 b Fv([0)p Fq(;)17 b Fv(1])32 b Fg(suc)m(h)i(that)e(for)h(all)d Fv(\003)i Fg(in)g Fr(S)3066 5009 y Fm(b)3101 4994 y Fg(,)270 5188 y Fh(\(a\))82 b Fg(F)-8 b(or)31 b(eac)m(h)j Fq(A)28 b Fr(2)g(F)1155 5203 y Fl(\024)p Fm(m)1272 5214 y Ff(\003)1321 5188 y Fq(;)45 b(f)1441 5203 y Fn(\003)1494 5188 y Fv(\()p Fq(A)28 b Fr(j)f(\001)17 b Fv(\))32 b Fg(is)g Fr(F)1973 5203 y Fn(\003)2022 5212 y Fe(\000)2078 5188 y Fg(-measurable.)262 5388 y Fh(\(b\))82 b Fg(F)-8 b(or)31 b(eac)m(h)j Fq(B)f Fr(2)28 b(F)1161 5403 y Fn(\003)1210 5412 y Fe(\000)1298 5388 y Fg(and)33 b Fq(!)e Fr(2)d Fv(\012)p Fq(;)45 b(f)1864 5403 y Fn(\003)1917 5388 y Fv(\()p Fq(B)33 b Fr(j)27 b Fq(!)t Fv(\))g(=)44 b Fx(1)-30 b Fv(1)2431 5403 y Fm(B)2491 5388 y Fv(\()p Fq(!)t Fv(\))p Fq(:)1852 5637 y Fv(5)p eop %%Page: 6 6 6 5 bop 275 573 a Fh(\(c\))81 b Fg(F)-8 b(or)31 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2959 y Fl(\000)p Fn(1])2225 2943 y Fv(.)257 3145 y Fh(2.16)48 b Fv(As)28 b(for)f(sp)s(eci\014cations,)i(in)e(the)h(sequel)h(w)m(e)g (shall)d(not)i(distinguish)e(notationally)478 3265 y(the)41 b(k)m(ernels)g Fq(f)1035 3280 y Fn(\003)1128 3265 y Fv(from)e(their)h (extensions)i(on)e Fr(F)2300 3280 y Fm(m)2362 3291 y Ff(\003)2439 3265 y Fr(\002)28 b(A)2624 3229 y Fp(Z)2713 3265 y Fr(!)40 b Fv([0)p Fq(;)17 b Fv(1])40 b(suc)m(h)h(that)478 3385 y Fq(f)526 3400 y Fn(\003)579 3385 y Fv(\()17 b Fr(\001)27 b(j)g Fq(!)t Fv(\))g(=)h(0)k(if)g Fq(!)42 b(=)-60 b Fr(2)28 b Fv(\012.)257 3584 y Fh(De\014nition)36 b(2.17)257 3705 y Fg(A)c(probabilit)m(y)e(measure)i Fq(\026)f Fg(on)h Fv(\(\012)p Fq(;)45 b Fr(F)10 b Fv(\))31 b Fg(is)g(said)g(to)g (b)s(e)38 b Fh(consisten)m(t)30 b Fg(with)i(a)f(LIS)h Fq(f)42 b Fg(if)257 3825 y(for)32 b(eac)m(h)i Fv(\003)27 b Fr(2)h(S)875 3840 y Fm(b)1379 3946 y Fq(\026)17 b(f)1503 3961 y Fn(\003)1611 3946 y Fv(=)56 b Fq(\026)97 b Fg(o)m(v)m(er)33 b Fr(F)2179 3961 y Fl(\024)p Fm(m)2296 3972 y Ff(\003)2346 3946 y Fq(:)873 b Fv(\(2.18\))257 4117 y Fg(The)34 b(family)c(of)i (these)i(measures)f(will)e(b)s(e)h(denoted)i Fr(G)6 b Fv(\()p Fq(f)11 b Fv(\))p Fg(.)257 4326 y Fh(Remarks)257 4464 y(2.19)48 b Fv(A)31 b Fs(Markov)i(chain)38 b Fv(of)30 b(range)h Fq(k)j Fv(is)c(a)h(measure)g(consisten)m(t)h(with)f(a)g(Mark) m(o)m(v)h(LIS)478 4585 y(of)g(range)h Fq(k)s Fv(.)257 4786 y Fh(2.20)48 b Fv(Measures)33 b(consisten)m(t)f(with)f(general,)g (non-necessarily)g(Mark)m(o)m(vian)g(LIS)h(w)m(ere)478 4907 y(initially)f(called)j Fs(Chains)i(with)g(c)-5 b(omplete)37 b(c)-5 b(onne)g(ctions)41 b Fv(b)m(y)36 b(Onicescu)g(and)f(Mi-)478 5027 y(ho)s(c)50 b(\(1935\).)95 b(These)51 b(ob)5 b(jects)52 b(ha)m(v)m(e)f(b)s(een)g(rein)m(tro)s(duced)f(sev)m(eral)h(times)e(in) 478 5147 y(the)37 b(literature)e(under)i(a)f(v)-5 b(ariet)m(y)36 b(of)g(names:)50 b Fs(chains)38 b(of)g(in\014nite)f(or)-5 b(der)47 b Fv(\(Har-)478 5268 y(ris,)41 b(1955\),)f Fq(g)t Fs(-me)-5 b(asur)g(es)46 b Fv(\(Keane,)41 b(1972\),)f Fs(uniform)h(martingales)46 b Fv(\(=random)478 5388 y(Mark)m(o)m(v)34 b(pro)s(cesses\))h(\(Kalik)m(o)m(w,)c(1990\),)h(.)16 b(.)g(.)g(.)1852 5637 y(6)p eop %%Page: 7 7 7 6 bop 404 573 a Fv(Finally)-8 b(,)29 b(w)m(e)j(in)m(tro)s(duce)f(a)g (strong)g(notion)f(of)h(uniqueness)i(needed)g(in)e(the)g(sequel.)257 853 y Fh(De\014nition)36 b(2.21)313 973 y(1\))81 b Fg(A)23 b(sp)s(eci\014cation)f Fq(\015)28 b Fg(satis\014es)23 b(a)g Fh(hereditary)j(uniqueness)h(condition)22 b Fg(\(HUC\))494 1094 y Fh(for)36 b(a)i(family)30 b Fr(H)j Fg(of)f(subsets)i(of)d Fo(Z)f Fg(if)h(for)g(all)f(\(p)s(ossibly)i(in\014nite\))f(sets)i Fq(V)49 b Fr(2)28 b(H)494 1214 y Fg(and)k(all)f(con\014gurations)h Fq(!)f Fr(2)d Fv(\012)p Fg(,)33 b(the)g(sp)s(eci\014cation)f Fq(\015)2544 1178 y Fn(\()p Fm(V)t(;!)r Fn(\))2747 1214 y Fg(de\014ned)i(b)m(y)644 1393 y Fq(\015)700 1342 y Fn(\()p Fm(V)t(;!)r Fn(\))695 1420 y(\003)870 1393 y Fv(\()17 b Fr(\001)26 b(j)i Fq(\030)5 b Fv(\))27 b(=)g Fq(\015)1302 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(functions)g(asso)s(ciated)f(to)h(LIS)g(singletons.)42 b(F)-8 b(or)30 b(a)h(LIS)g Fq(f)42 b Fv(and)30 b(a)257 3393 y(con\014guration)i Fq(!)f Fr(2)d Fv(\012,)33 b(let)1368 3572 y Fq(f)1416 3587 y Fm(i)1444 3572 y Fv(\()p Fq(!)t Fv(\))55 b Fk(,)g Fq(f)1821 3587 y Fl(f)p Fm(i)p Fl(g)1919 3572 y Fv(\()p Fq(!)2018 3587 y Fm(i)2074 3572 y Fr(j)27 b Fq(!)2194 3531 y Fm(i)p Fl(\000)p Fn(1)2190 3596 y Fl(\0001)2320 3572 y Fv(\))p Fq(:)910 b Fv(\(3.1\))257 3751 y(In)45 b(the)g(shift-in)m(v)-5 b(arian)m(t)43 b(case,)48 b(the)d(function)g Fq(f)2084 3766 y Fn(0)2168 3751 y Fv(is)f(a)g Fq(g)t Fv(-function)f(in)h(the)h(sense)i(of)257 3871 y(Keane)33 b(\(1972\).)404 3991 y(The)e(follo)m(wing)d(theorem)i (expresses)k(the)d(equiv)-5 b(alence)31 b(b)s(et)m(w)m(een)i(the)e (description)257 4112 y(in)f(terms)h(of)g(LIS)g(and)g(the)g(usual)f (description)h(in)f(terms)g(of)h(transition)e(probabilities)257 4232 y(\(=LIS)k(singletons\).)257 4422 y Fh(Theorem)k(3.2)h 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5238 y Fm(i)2278 5223 y Fv(\))55 b(=)g(1)p Fq(:)717 b Fv(\(3.3\))1852 5637 y(7)p eop %%Page: 8 8 8 7 bop 257 573 a Fg(Then)35 b(there)f(exists)h(a)e(unique)h(left)f(in) m(terv)-5 b(al-sp)s(eci\014cation)31 b Fq(f)40 b Fk(,)29 b Fv(\()p Fq(f)2796 588 y Fn(\003)2850 573 y Fv(\))2888 588 y Fn(\003)p Fl(2S)3027 600 y Fi(b)3095 573 y Fg(suc)m(h)35 b(that)257 693 y Fq(f)305 708 y Fm(i)361 693 y Fv(=)28 b Fq(g)512 708 y Fm(i)540 693 y Fq(;)49 b Fg(for)32 b(all)e Fq(i)j Fg(in)f Fo(Z)p Fg(.)41 b(F)-8 b(urthermore:)294 897 y Fh(\(i\))80 b Fq(f)43 b Fg(satis\014es)33 b(\(in)f(fact,)g(it)g (is)g(de\014ned)i(b)m(y\))f(the)g(prop)s(ert)m(y)1334 1117 y Fq(f)1382 1132 y Fn([)p Fm(l)q(;m)p Fn(])1557 1117 y Fv(=)28 b Fq(f)1709 1132 y Fn([)p Fm(l)q(;n)p Fn(])1836 1117 y Fq(f)1884 1132 y Fn([)p Fm(n)p Fn(+1)p Fm(;m)p Fn(])2273 1117 y Fg(o)m(v)m(er)33 b Fr(F)2553 1132 y Fl(\024)p Fm(n)3295 1117 y Fv(\(3.4\))494 1337 y Fg(for)f(eac)m(h)h Fq(l)r(;)17 b(m;)g(n)28 b Fr(2)g Fo(Z)d Fv(:)j Fq(l)i Fr(\024)e Fq(n)g(<)f(m)p Fg(.)262 1540 y Fh(\(ii\))80 b Fq(f)41 b Fg(is)30 b(non-n)m(ull)f(on)h Fv(\012)h Fg(if,)f(and)h(only)f(if,)f(so)i(are)g(the)g(functions)f Fq(g)2832 1555 y Fm(i)2860 1540 y Fg(,)h(that)f(is,)h(if)e(and)494 1660 y(only)j(if)f Fq(g)844 1675 y Fm(i)872 1660 y Fv(\()p Fq(!)t Fv(\))c Fq(>)g Fv(0)33 b Fg(for)f(eac)m(h)h Fq(i)28 b Fr(2)g Fo(Z)i Fg(and)j(eac)m(h)g Fq(!)e Fr(2)d Fv(\012)2512 1624 y Fm(i)2512 1685 y Fl(\0001)2642 1660 y Fg(.)249 1864 y Fh(\(iii\))79 b Fq(f)37 b Fg(is)26 b(w)m(eakly)h(non-n)m(ull)e (on)h Fv(\012)h Fg(if,)g(and)f(only)g(if,)h(so)f(are)h(the)f(functions) h Fq(g)3115 1879 y Fm(i)3143 1864 y Fg(,)g(that)g(is,)494 1984 y(if)22 b(and)i(only)f(if)g(for)g(eac)m(h)i Fq(i)j Fr(2)g Fo(Z)21 b Fg(there)k(exists)f Fq(\033)2193 1999 y Fm(i)2250 1984 y Fr(2)k Fv(\012)2414 2000 y Fl(f)p Fm(i)p Fl(g)2537 1984 y Fg(suc)m(h)d(that)e Fq(g)2997 1999 y Fm(i)3025 1984 y Fv(\()p Fq(\033)3118 1999 y Fm(i)3163 1984 y Fq(!)3228 1943 y Fm(i)p Fl(\000)p Fn(1)3224 2006 y Fl(\0001)3354 1984 y Fv(\))k Fq(>)494 2105 y Fv(0)32 b Fg(for)g(all)e Fq(!)924 2063 y Fm(i)p Fl(\000)p Fn(1)920 2126 y Fl(\0001)1077 2105 y Fr(2)f Fv(\012)1242 2063 y Fm(i)p Fl(\000)p Fn(1)1242 2126 y Fl(\0001)1404 2105 y Fg(suc)m(h)34 b(that)e Fq(\033)1890 2120 y Fm(i)1936 2105 y Fq(!)2001 2063 y Fm(i)p Fl(\000)p Fn(1)1997 2126 y Fl(\0001)2154 2105 y Fr(2)c Fv(\012)2318 2068 y Fm(i)2318 2129 y Fl(\0001)2448 2105 y Fg(.)249 2308 y Fh(\(iv\))80 b Fr(G)6 b Fv(\()p Fq(f)11 b Fv(\))28 b(=)f Fr(f)p Fq(\026)h Fv(:)f Fq(\026f)1137 2323 y Fm(i)1193 2308 y Fv(=)h Fq(\026;)48 b Fg(for)33 b(all)d Fq(i)j Fg(in)f Fo(Z)p Fr(g)p Fg(.)257 2511 y Fv(The)40 b(pro)s(of)d(of)h(this)g(result)g(is)f(rather)i (simple)d(\(it)i(is)f(sp)s(elled)h(up)h(in)e(F)-8 b(ern\023)-49 b(andez)39 b(and)257 2632 y(Maillard,)30 b(2003\).)43 b(The)33 b(follo)m(wing)c(theorem)k(is)e(the)i(analogous)e(result)h (for)g(sp)s(eci\014ca-)257 2752 y(tions.)53 b(Let)36 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Fj(o)2884 3488 y Fq(:)335 b Fv(\(3.11\))249 3777 y Fh(\(iii\))79 b Fg(The)32 b Fh(in)m(terdep)s(endence)e(co)s(e\016cien)m(ts)25 b Fg(for)g(a)h(family)d(of)i(probabilit)m(y)f(k)m(ernels)494 3897 y Fq(\031)31 b Fv(=)684 3817 y Fj(\000)729 3897 y Fq(\031)784 3913 y Fl(f)p Fm(i)p Fl(g)883 3817 y Fj(\001)929 3936 y Fm(i)p Fl(2)p Fp(Z)1065 3897 y Fq(;)45 b(\031)1192 3913 y Fl(f)p Fm(i)p Fl(g)1318 3897 y Fv(:)28 b Fr(F)1445 3912 y Fm(i)1495 3897 y Fr(\002)23 b Fv(\012)28 b Fr(!)f Fv([0)p Fq(;)17 b Fv(1])32 b Fg(are)h(de\014ned)h(b)m(y)1119 4163 y Fq(C)1189 4178 y Fm(ij)1249 4163 y Fv(\()p Fq(\031)t Fv(\))55 b Fk(,)76 b Fv(sup)1572 4245 y Fm(\030)s(;\021)r Fl(2)p Fn(\012)1594 4352 y Fm(\030)1628 4310 y Fe(6)p Ff(=)p Fi(j)1637 4352 y Fn(=)10 b Fm(\021)1778 4049 y Fj(\015)1778 4108 y(\015)1778 4168 y(\015)1847 4097 y Fl(\016)1833 4160 y Fq(\031)1892 4179 y Fl(f)p Fm(i)p Fl(g)2019 4163 y Fv(\()17 b Fr(\001)26 b(j)i Fq(\030)5 b Fv(\))p Fr(\000)2388 4097 y Fl(\016)2374 4160 y Fq(\031)2433 4179 y Fl(f)p Fm(i)p Fl(g)2560 4163 y Fv(\()17 b Fr(\001)27 b(j)g Fq(\021)t Fv(\))2815 4049 y Fj(\015)2815 4108 y(\015)2815 4168 y(\015)3246 4163 y Fv(\(3.12\))494 4589 y Fg(for)22 b(all)f Fq(i;)c(j)34 b Fr(2)28 b Fo(Z)p Fg(.)38 b(Here)24 b(w)m(e)g(use)g(the)g(v)-5 b(ariation)21 b(norm)h(and)2650 4523 y Fl(\016)2636 4586 y Fq(\031)2695 4605 y Fl(f)p Fm(i)p Fl(g)2817 4589 y Fg(is)h(the)g(pro)5 b(jection)494 4736 y(of)33 b Fq(\031)661 4752 y Fl(f)p Fm(i)p Fl(g)794 4736 y Fg(o)m(v)m(er)i Fr(f)p Fq(i)p Fr(g)e Fg(that)h(is)1496 4670 y Fl(\016)1482 4733 y Fq(\031)1541 4752 y Fl(f)p Fm(i)p Fl(g)1670 4736 y Fv(\()p Fq(A)28 b Fr(j)f Fq(!)t Fv(\))i Fk(,)g Fq(\031)2158 4752 y Fl(f)p Fm(i)p Fl(g)2274 4736 y Fv(\()p Fr(f)p Fq(\033)2417 4751 y Fm(i)2473 4736 y Fr(2)f Fq(A)p Fr(g)g(j)f Fq(!)t Fv(\))33 b Fg(for)g(all)f Fq(A)e Fr(2)g(F)3467 4751 y Fm(i)494 4857 y Fg(and)i Fq(!)f Fr(2)d Fv(\012)p Fg(.)257 5038 y Fh(Theorem)37 b(3.13)257 5158 y Fg(A)g(LIS)f Fq(f)47 b Fg(on)36 b Fv(\(\012)p Fq(;)17 b Fr(F)10 b Fv(\))35 b Fg(satis\014es)i(a)f(HUC)g(if)f(it)g (satis\014es)i(one)g(of)e(the)i(follo)m(wing)c(condi-)257 5279 y(tions:)1852 5637 y Fv(9)p eop %%Page: 10 10 10 9 bop 313 636 a Fh(1\))81 b(Harris)36 b Fg(\(1955\):)52 b(F)-8 b(or)37 b(eac)m(h)i Fq(i)28 b Fr(2)g Fo(Z)p Fq(;)1888 542 y Fj(X)1894 752 y Fm(n)p Fl(\025)p Fn(1)2092 512 y Fm(n)2050 542 y Fj(Y)2049 754 y Fm(k)r Fn(=1)2194 496 y Fj(\022)2268 636 y Fv(1)22 b Fr(\000)2448 569 y(jAj)p 2448 613 135 4 v 2491 705 a Fv(2)2593 636 y(v)-5 b(ar)2726 651 y Fm(k)2768 636 y Fv(\()p Fq(f)2854 651 y Fm(i)2883 636 y Fv(\))2921 496 y Fj(\023)3022 636 y Fv(=)27 b(+)p Fr(1)37 b Fg(and)494 851 y Fq(f)43 b Fg(is)32 b(w)m(eakly)h(non-n)m (ull)e(on)i Fv(\012)p Fg(.)313 1066 y Fh(2\))81 b(Hulse)31 b Fg(\(1997\):)42 b(F)-8 b(or)31 b(eac)m(h)i Fq(i)28 b Fr(2)g Fo(Z)p Fq(;)1829 971 y Fj(X)1837 1184 y Fm(k)r Fl(\025)p Fn(0)1990 1066 y Fq(\016)2033 1081 y Fm(i)p Fl(\000)p Fm(k)2171 1066 y Fv(\()p Fq(f)2257 1081 y Fm(i)2285 1066 y Fv(\))g Fq(<)f Fv(+)p Fr(1)32 b Fg(and)g Fq(f)43 b Fg(is)31 b(con)m(tin)m(uous)494 1282 y(and)h(non-n)m(ull)f(on)i Fv(\012)p Fg(.)313 1486 y Fh(3\))81 b(One-sided)37 b(Dobrushin)p Fg(:)44 b(F)-8 b(or)31 b(eac)m(h)i Fq(i)28 b Fr(2)g Fo(Z)p Fq(;)2285 1391 y Fj(X)2301 1601 y Fm(j)t()28 b Fv(0)g Fg(so)494 2024 y(that)38 b(for)h(ev)m(ery)i(cylinder)d (set)i Fq(A)f Fv(=)f Fr(f)p Fq(x)1998 1988 y Fm(m)1998 2050 y(l)2065 2024 y Fr(g)h(2)g Fv(\012)2329 1988 y Fm(m)2329 2050 y(l)2435 2024 y Fg(there)g(exists)h(an)f(in)m(teger)g Fq(n)494 2144 y Fg(suc)m(h)34 b(that)866 2364 y Fq(f)914 2380 y Fn([)p Fm(n;m)p Fn(])1082 2364 y Fv(\()p Fq(A)28 b Fr(j)g Fq(\030)5 b Fv(\))27 b Fr(\025)h Fq(K)23 b(f)1649 2380 y Fn([)p Fm(n;m)p Fn(])1818 2364 y Fv(\()p Fq(A)k Fr(j)h Fq(\021)t Fv(\))97 b Fg(for)32 b(all)f Fq(\030)5 b(;)17 b(\021)30 b Fr(2)e Fv(\012)g Fq(:)373 b Fv(\(3.14\))404 2626 y(The)42 b(last)e(t)m(w)m(o)i(conditions,)g(pro)m(v)m(en)g(in)f(a) f(companion)g(pap)s(er)h(\(F)-8 b(ern\023)-49 b(andez)42 b(and)257 2746 y(Maillard,)33 b(2003\),)h(are)h(in)f(fact)g (adaptations)g(of)g(the)h(follo)m(wing)d(w)m(ell)i(kno)m(wn)i(criteria) 257 2867 y(for)c(sp)s(eci\014cations.)257 3063 y Fh(Theorem)37 b(3.15)257 3184 y Fg(A)31 b(sp)s(eci\014cation)f Fq(\015)36 b Fg(on)31 b Fv(\(\012)p Fq(;)17 b Fr(F)10 b Fv(\))30 b Fg(satis\014es)i(a)e(HUC)h(if)f(it)g(satis\014es)h(one)g(of)g(the)g (follo)m(wing)257 3304 y(conditions:)313 3507 y Fh(1\))81 b(Dobrushin)36 b Fg(\(1968\),)g(Lanford)f(\(1973\):)50 b Fv(sup)2112 3589 y Fm(i)p Fl(2)p Fp(Z)2259 3413 y Fj(X)2269 3624 y Fm(j)t Fl(2)p Fp(Z)2436 3507 y Fq(C)2506 3522 y Fm(ij)2567 3507 y Fv(\()p Fq(\015)5 b Fv(\))27 b Fq(<)h Fv(1)36 b Fg(and)g Fq(\015)41 b Fg(con)m(tin-)494 3699 y(uous.)313 3902 y Fh(2\))81 b(Georgii)30 b Fg(\(1974\))i Fh(b)s(oundary-uniformit)m(y)p Fg(:)42 b(There)33 b(exists)g(a)e (constan)m(t)i Fq(K)h(>)494 4022 y Fv(0)e Fg(so)h(that)f(for)g(ev)m (ery)i(cylinder)f(set)g Fq(A)28 b Fr(2)g(F)42 b Fg(there)33 b(exists)h Fv(\003)27 b Fr(2)h(S)2913 4037 y Fm(b)2980 4022 y Fg(suc)m(h)34 b(that)1103 4242 y Fq(\015)1154 4257 y Fn(\003)1207 4242 y Fv(\()p Fq(A)28 b Fr(j)f Fq(\030)5 b Fv(\))27 b Fr(\025)h Fq(K)c(\015)1777 4257 y Fn(\003)1830 4242 y Fv(\()p Fq(A)k Fr(j)f Fq(\021)t Fv(\))97 b Fg(for)32 b(all)f Fq(\030)5 b(;)17 b(\021)30 b Fr(2)e Fv(\012)g Fq(:)257 4504 y Fv(W)-8 b(e)33 b(remark)g(that)f(the)h(conditions)f(in) m(v)m(olv)m(e)g(no)h(non-n)m(ullness)g(assumption.)257 4837 y Fw(4)161 b(Main)55 b(results)257 5056 y Fv(F)-8 b(or)32 b(a)g(LIS)h Fq(f)44 b Fv(on)32 b(\012)h(let)f(us)h(denote,)g (for)g(eac)m(h)g(\003)27 b Fr(2)h(S)7 b Fq(;)46 b(k)30 b Fr(\025)f Fq(m)2603 5071 y Fn(\003)2689 5056 y Fv(and)j Fq(!)f Fr(2)d Fv(\012)3134 5020 y Fm(k)3134 5080 y Fl(\0001)3264 5056 y Fv(,)672 5286 y Fq(c)714 5245 y Fm(!)714 5311 y Fn(\003)767 5286 y Fv(\()p Fq(f)853 5301 y Fm(k)895 5286 y Fv(\))56 b Fk(,)110 b Fv(inf)1122 5348 y Fm(\033)1162 5359 y Ff(\003)1207 5348 y Fl(2)p Fn(\012)1305 5359 y Ff(\003)1368 5206 y Fj(\010)1426 5286 y Fq(f)1474 5301 y Fm(k)1533 5206 y Fj(\000)1579 5286 y Fq(\033)1634 5301 y Fn(\003)1704 5286 y Fq(!)1765 5302 y Fn(])p Fl(\0001)p Fm(;k)r Fn(])p Fl(n)p Fn(\003)2076 5206 y Fj(\001)2149 5286 y Fv(:)28 b Fq(\033)2259 5301 y Fn(\003)2329 5286 y Fq(!)2390 5302 y Fn(])p Fl(\0001)p Fm(;k)r Fn(])p Fl(n)p Fn(\003)2729 5286 y Fr(2)g Fv(\012)2893 5245 y Fm(k)2893 5311 y Fl(\0001)3023 5206 y Fj(\011)3295 5286 y Fv(\(4.1\))1828 5637 y(10)p eop %%Page: 11 11 11 10 bop 257 573 a Fv(and)419 748 y Fq(\016)466 707 y Fm(!)462 773 y Fn(\003)516 748 y Fv(\()p Fq(f)602 763 y Fm(k)645 748 y Fv(\))55 b Fk(,)871 653 y Fj(X)879 865 y Fm(j)t Fl(2)p Fn(\003)1032 748 y Fv(sup)1195 637 y Fj(n)1261 663 y(\014)1261 723 y(\014)1295 748 y Fq(f)1343 763 y Fm(k)1402 667 y Fj(\000)1448 748 y Fq(!)1513 707 y Fm(k)1509 773 y Fl(\0001)1638 667 y Fj(\001)1706 748 y Fr(\000)22 b Fq(f)1853 763 y Fm(k)1913 667 y Fj(\000)1958 748 y Fq(\033)2017 707 y Fm(k)2013 773 y Fl(\0001)2144 667 y Fj(\001)2189 663 y(\014)2189 723 y(\014)2250 748 y Fv(:)28 b Fq(\033)k Fr(2)c Fv(\012)2556 707 y Fm(k)2556 773 y Fl(\0001)2686 748 y Fq(;)44 b(\033)2844 688 y Fl(6)p Fn(=)p Fm(j)2849 748 y Fv(=)34 b Fq(!)3024 637 y Fj(o)3106 748 y Fq(:)162 b Fv(\(4.2\))257 1020 y(Similarly)-8 b(,)30 b(for)k(a)f(sp)s(eci\014cation)g Fq(\015)39 b Fv(on)34 b(\012,)h(let)e(us)h(denote,)h(for)e(eac)m(h)i Fq(!)e Fr(2)d Fv(\012)k(and)g(eac)m(h)257 1140 y Fq(k)s(;)17 b(j)34 b Fr(2)28 b Fo(Z)905 1261 y Fq(c)947 1220 y Fm(!)947 1285 y(j)998 1261 y Fv(\()p Fq(\015)1087 1276 y Fm(k)1129 1261 y Fv(\))55 b Fk(,)107 b Fv(min)1355 1323 y Fm(\033)1395 1333 y Fi(j)1428 1323 y Fl(2)p Fn(\012)1526 1337 y Fe(f)p Fi(j)s Fe(g)1639 1180 y Fj(\010)1697 1261 y Fq(\015)1748 1276 y Fm(k)1807 1180 y Fj(\000)1852 1261 y Fq(\033)1907 1276 y Fm(j)1961 1261 y Fq(!)2022 1276 y Fl(f)p Fm(j)t Fl(g)2125 1257 y Fi(c)2161 1180 y Fj(\001)2234 1261 y Fv(:)28 b Fq(\033)2344 1276 y Fm(j)2397 1261 y Fq(!)2458 1276 y Fl(f)p Fm(j)t Fl(g)2561 1257 y Fi(c)2625 1261 y Fr(2)g Fv(\012)2789 1180 y Fj(\011)3295 1261 y Fv(\(4.3\))257 1472 y(and)802 1592 y Fq(\016)849 1551 y Fm(!)845 1617 y(j)916 1592 y Fv(\()p Fq(\015)1005 1607 y Fm(k)1047 1592 y Fv(\))55 b Fk(,)g Fv(sup)1437 1482 y Fj(n)1503 1592 y Fr(j)p Fq(\015)1582 1607 y Fm(k)1624 1592 y Fv(\()p Fq(!)t Fv(\))21 b Fr(\000)i Fq(\015)1937 1607 y Fm(k)1979 1592 y Fv(\()p Fq(\033)t Fv(\))p Fr(j)28 b Fv(:)f Fq(\033)32 b Fr(2)c Fv(\012)p Fq(;)45 b(\033)2634 1532 y Fl(6)p Fn(=)p Fm(j)2639 1592 y Fv(=)34 b Fq(!)2814 1482 y Fj(o)2924 1592 y Fq(:)344 b Fv(\(4.4\))257 1815 y Fh(De\014nition)36 b(4.5)294 1935 y(\(i\))80 b Fg(A)41 b(LIS)g Fq(f)51 b Fg(on)41 b Fv(\012)g Fg(is)g(said)f(to)h(ha)m(v)m(e)h(a)e Fh(go)s(o)s(d)48 b(future)f(\(GF\))41 b Fg(if)e(it)h(is)h(non-n)m(ull) 494 2064 y(on)36 b Fv(\012)h Fg(and)f(for)g(eac)m(h)h Fv(\003)d Fr(2)g(S)7 b Fg(,)39 b(there)e(exists)g(a)f(sequence)2663 1984 y Fj(\010)2721 2064 y Fq(")2767 2028 y Fn(\003)2767 2090 y Fm(k)2820 1984 y Fj(\011)2878 2103 y Fm(k)r Fl(2)p Fp(N)3052 2064 y Fg(of)g(p)s(ositiv)m(e)494 2198 y(n)m(um)m(b)s(ers)d (suc)m(h)h(that)1318 2123 y Fj(P)1424 2227 y Fm(k)1483 2198 y Fq(")1529 2161 y Fn(\003)1529 2223 y Fm(k)1609 2198 y Fq(<)28 b Fv(+)p Fr(1)k Fg(for)g(whic)m(h)1443 2373 y Fv(sup)1387 2466 y Fm(!)r Fl(2)p Fn(\012)1531 2443 y Fi(k)1531 2487 y Fe(\0001)1678 2373 y Fq(c)1720 2332 y Fm(!)1720 2397 y Fn(\003)1773 2373 y Fv(\()p Fq(f)1859 2388 y Fm(k)1902 2373 y Fv(\))1940 2332 y Fl(\000)p Fn(1)2051 2373 y Fq(\016)2098 2332 y Fm(!)2094 2397 y Fn(\003)2148 2373 y Fv(\()p Fq(f)2234 2388 y Fm(k)2277 2373 y Fv(\))55 b Fr(\024)h Fq(")2549 2332 y Fn(\003)2549 2397 y Fm(k)3295 2373 y Fv(\(4.6\))494 2636 y Fg(for)32 b(eac)m(h)h Fq(k)e Fr(\025)d Fq(m)1134 2651 y Fn(\003)1187 2636 y Fg(.)262 2827 y Fh(\(ii\))80 b Fg(A)31 b(LIS)g Fq(f)42 b Fg(on)31 b Fv(\012)h Fg(is)e(said)h(to)g(ha)m(v)m(e)h(an)f Fh(exp)s(onen)m (tially-go)s(o)s(d)36 b(future)g(\(EGF\))494 2947 y Fg(if)31 b(it)h(is)g(non-n)m(ull)f(on)h Fv(\012)h Fg(and)g(there)g(exists)g(a)g (real)e Fq(a)d(>)g Fv(1)k Fg(suc)m(h)i(that)1096 3122 y Fv(lim)17 b(sup)1156 3205 y Fm(k)r Fl(!1)1428 3122 y Fq(a)1479 3081 y Fl(j)p Fm(k)r Fl(\000)p Fm(j)t Fl(j)1737 3122 y Fv(sup)1682 3216 y Fm(!)r Fl(2)p Fn(\012)1826 3193 y Fi(k)1826 3237 y Fe(\0001)1972 3122 y Fq(c)2014 3081 y Fm(!)2014 3147 y(j)2065 3122 y Fv(\()p Fq(f)2151 3137 y Fm(k)2194 3122 y Fv(\))2232 3081 y Fl(\000)p Fn(1)2342 3122 y Fq(\016)2389 3081 y Fm(!)2385 3147 y(j)2440 3122 y Fv(\()p Fq(f)2526 3137 y Fm(k)2568 3122 y Fv(\))56 b Fq(<)f Fr(1)402 b Fv(\(4.7\))494 3386 y Fg(for)32 b(all)e Fq(j)k Fr(2)28 b Fo(Z)p Fg(.)249 3577 y Fh(\(iii\))79 b Fg(A)28 b(sp)s(eci\014cation)f Fq(\015)34 b Fg(on)28 b Fv(\012)g Fg(is)g(said)f(to)h(ha)m(v)m(e)i(an)e Fh(exp)s(onen)m (tially-go)s(o)s(d)k(future)494 3697 y(\(EGF\))f Fg(if)h(it)f(is)h (non-n)m(ull)g(on)g Fv(\012)h Fg(and)g(there)g(exists)g(a)g(real)e Fq(a)d(>)g Fv(1)k Fg(suc)m(h)i(that)1130 3872 y Fv(lim)17 b(sup)1189 3954 y Fm(k)r Fl(!1)1462 3872 y Fq(a)1513 3831 y Fl(j)p Fm(k)r Fl(\000)p Fm(j)t Fl(j)1726 3872 y Fv(sup)1728 3953 y Fm(!)r Fl(2)p Fn(\012)1917 3872 y Fq(c)1959 3831 y Fm(!)1959 3897 y(j)2010 3872 y Fv(\()p Fq(\015)2099 3887 y Fm(k)2141 3872 y Fv(\))2179 3831 y Fl(\000)p Fn(1)2290 3872 y Fq(\016)2337 3831 y Fm(!)2333 3897 y(j)2404 3872 y Fv(\()p Fq(\015)2493 3887 y Fm(k)2535 3872 y Fv(\))55 b Fq(<)h Fr(1)435 b Fv(\(4.8\))494 4100 y Fg(for)32 b(all)e Fq(j)k Fr(2)28 b Fo(Z)p Fg(.)257 4277 y Fh(De\014nition)36 b(4.9)257 4398 y Fg(Let)d(us)g(in)m(tro)s (duce)g(the)g(follo)m(wing)d(sets.)672 4570 y Fv(\002)e Fk(,)f Fr(f)p Fg(LIS)33 b Fq(f)11 b Fg(con)m(tin)m(uous)33 b(and)f(non-n)m(ull)g(on)g Fv(\012)p Fr(g)672 4691 y Fv(\005)c Fk(,)f Fr(f)p Fg(sp)s(eci\014cations)33 b Fq(\015)k Fg(con)m(tin)m(uous)d(and)e(non-n)m(ull)f(on)i Fv(\012)p Fr(g)p Fg(,)672 4811 y Fv(\002)748 4826 y Fn(1)815 4811 y Fk(,)28 b Fr(f)o Fq(f)39 b Fr(2)28 b Fv(\002)f(:)h Fq(f)43 b Fg(has)33 b(a)g(GF)o Fr(g)p Fg(,)672 4932 y Fv(\005)745 4947 y Fn(1)813 4932 y Fk(,)27 b Fr(f)p Fq(\015)32 b Fr(2)c Fv(\005)g(:)g Fr(jG)6 b Fv(\()p Fq(\015)f Fv(\))p Fr(j)27 b Fv(=)h(1)p Fr(g)o Fg(,)672 5052 y Fv(\005)745 5067 y Fn(2)813 5052 y Fk(,)f Fr(f)p Fq(\015)32 b Fr(2)c Fv(\005)g(:)g Fq(\015)38 b Fg(satis\014es)33 b(a)f(HUC)h(o)m(v)m(er)h (all)c Fv([)p Fq(i;)17 b Fv(+)p Fr(1)p Fv([)p Fq(;)45 b(i)27 b Fr(2)i Fo(Z)p Fr(g)m Fg(,)672 5172 y Fv(\002)748 5187 y Fn(2)815 5172 y Fk(,)f Fr(f)o Fq(f)39 b Fr(2)28 b Fv(\002)f(:)h Fq(f)43 b Fg(satis\014es)34 b(a)e(HUC)q Fr(g)p Fg(,)672 5293 y Fv(\002)748 5308 y Fn(3)815 5293 y Fk(,)c Fr(f)o Fq(f)39 b Fr(2)28 b Fv(\002)f(:)h Fq(f)43 b Fg(has)33 b(an)g(EGF)o Fr(g)p Fg(,)672 5413 y Fv(\005)745 5428 y Fn(3)813 5413 y Fk(,)27 b Fr(f)p Fq(\015)32 b Fr(2)c Fv(\005)g(:)g Fq(\015)38 b Fg(has)32 b(an)h(EGF)p Fr(g)o Fg(.)1828 5637 y Fv(11)p eop %%Page: 12 12 12 11 bop 404 573 a Fv(W)-8 b(e)33 b(remark)f(that)h(eac)m(h)h(of)e (the)h(LIS)g(or)g(sp)s(eci\014cations)f(of)h(an)m(y)g(of)g(the)g (preceding)257 693 y(sets)45 b(has)f(at)f(least)g(one)g(consisten)m(t)i (measure.)76 b(This)43 b(is)g(b)s(ecause)i(the)f(\(in)m(teresting)257 814 y(part)31 b(of)7 b(\))30 b(the)h(con\014guration)f(space)i(is)e (compact)h(and)g(the)g(LIS)g(or)f(sp)s(eci\014cations)h(are)257 934 y(assumed)f(to)f(b)s(e)h(con)m(tin)m(uous.)43 b(Indeed,)31 b(as)f(the)g(space)g(of)f(probabilit)m(y)e(measures)j(on)f(a)257 1054 y(compact)k(space)i(is)e(w)m(eakly)h(compact,)f(ev)m(ery)j (sequence)g(of)d(measures)h Fq(\015)2996 1069 y Fn(\003)3045 1077 y Fi(n)3091 1054 y Fv(\()17 b Fr(\001)45 b(j)28 b Fq(!)3340 1018 y Fl(f)p Fm(n)p Fl(g)3457 1054 y Fv(\))257 1175 y(or)k Fq(f)424 1190 y Fn(\003)473 1198 y Fi(n)520 1175 y Fv(\()17 b Fr(\001)43 b(j)28 b Fq(!)767 1139 y Fl(f)p Fm(n)p Fl(g)883 1175 y Fv(\),)k(for)g(\(\003)1235 1190 y Fm(n)1282 1175 y Fv(\))f(an)h(exhausting)h(sequence)h(of)e (regions)f(and)i(\()p Fq(!)3114 1139 y Fl(f)p Fm(n)p Fl(g)3230 1175 y Fv(\))f(a)g(se-)257 1295 y(quence)24 b(of)d(con\014gurations,)j(has)e(a)g(w)m(eakly)g(con)m(v)m(ergen)m(t)i (subsequence.)44 b(By)22 b(con)m(tin)m(uit)m(y)257 1416 y(of)39 b(the)h(transitions)e(the)h(limit)d(is)i(resp)s(ectiv)m(ely)j (a)d(Gibbs)h(measure)g(or)g(a)g(consisten)m(t)257 1536 y(c)m(hain.)404 1705 y(Consider)33 b(the)g(function)1037 1968 y Fq(F)1100 1983 y Fn(\003)p Fm(;n)1232 1968 y Fv(\()p Fq(!)1331 1983 y Fn(\003)1412 1968 y Fr(j)27 b Fq(!)t Fv(\))54 b Fk(,)1882 1893 y Fq(f)1930 1909 y Fn([)p Fm(l)1971 1920 y Ff(\003)2016 1909 y Fm(;n)p Fn(])2119 1813 y Fj(\000)2165 1893 y Fq(!)2230 1852 y Fm(n)2226 1918 y(l)2247 1929 y Ff(\003)2324 1893 y Fr(j)27 b Fq(!)2440 1908 y Fn(\003)2489 1917 y Fe(\000)2545 1813 y Fj(\001)p 1767 1945 939 4 v 1767 2042 a Fq(f)1815 2057 y Fn([)p Fm(l)1856 2068 y Ff(\003)1902 2057 y Fm(;n)p Fn(])2004 1961 y Fj(\000)2050 2042 y Fq(!)2111 2057 y Fn(\003)2160 2038 y Fi(c)2192 2057 y Fl(\\)p Fn([)p Fm(l)2280 2068 y Ff(\003)2325 2057 y Fm(;n)p Fn(])2439 2042 y Fr(j)g Fq(!)2555 2057 y Fn(\003)2604 2066 y Fe(\000)2660 1961 y Fj(\001)3246 1968 y Fv(\(4.10\))257 2250 y(for)41 b(all)f(\003)i Fr(2)h(S)7 b Fv(,)45 b Fq(n)e Fr(\025)g Fq(m)1224 2265 y Fn(\003)1319 2250 y Fv(and)e Fq(!)46 b Fr(2)d Fv(\012.)71 b(The)42 b(con)m(tin)m(uit)m(y)g(of)e Fq(f)53 b Fv(implies)39 b(that)i(the)257 2370 y(functions)29 b Fq(F)737 2385 y Fn(\003)p Fm(;n)853 2370 y Fv(\()p Fq(!)952 2385 y Fn(\003)1032 2370 y Fr(j)f(\001)17 b Fv(\))28 b(are)h(con)m(tin)m(uous)h(on)e(\012)2044 2385 y Fn(\003)2093 2366 y Fi(c)2159 2370 y Fv(for)g(eac)m(h)i Fq(!)2581 2385 y Fn(\003)2662 2370 y Fr(2)e Fv(\012)2826 2385 y Fn(\003)2879 2370 y Fv(.)43 b(W)-8 b(e)29 b(use)h(these)257 2490 y(functions)j(to)f(in)m(tro)s(duce)h(the)g(map)1436 2706 y Fq(b)28 b Fv(:)f(\002)1635 2721 y Fn(1)1702 2706 y Fr(!)h Fv(\005)g Fq(;)44 b(f)38 b Fr(7!)28 b Fq(\015)2272 2665 y Fm(f)257 2922 y Fv(de\014ned)34 b(b)m(y)1168 3042 y Fq(\015)1224 2995 y Fm(f)1219 3069 y Fn(\003)1288 3042 y Fv(\()p Fq(!)1387 3057 y Fn(\003)1468 3042 y Fr(j)27 b Fq(!)t Fv(\))55 b Fk(,)106 b Fv(lim)1813 3102 y Fm(n)p Fl(!)p Fn(+)p Fl(1)2069 3042 y Fq(F)2132 3057 y Fn(\003)p Fm(;n)2248 3042 y Fv(\()p Fq(!)2347 3057 y Fn(\003)2427 3042 y Fr(j)27 b Fq(!)t Fv(\))661 b(\(4.11\))257 3249 y(for)32 b(all)f(\003)c Fr(2)h(S)41 b Fv(and)32 b Fq(!)f Fr(2)d Fv(\012.)257 3434 y Fh(Theorem)37 b(4.12)h(\(LIS)f Fk( )g Fh(sp)s(eci\014cation\))313 3554 y(1\))81 b Fg(The)33 b(map)f Fq(b)h Fg(is)f(w)m(ell)g(de\014ned.)45 b(That)33 b(is,)f(for)g Fq(f)38 b Fr(2)28 b Fv(\002)2429 3569 y Fn(1)507 3756 y Fh(\(a\))81 b Fg(the)33 b(limit)c(\(4.11\))j(exists)h (for)f(all)f Fv(\003)c Fr(2)h(S)40 b Fg(and)33 b Fq(!)e Fr(2)d Fv(\012)p Fg(.)499 3917 y Fh(\(b\))81 b Fq(\015)786 3881 y Fm(f)864 3917 y Fg(is)32 b(a)g(sp)s(eci\014cation)g(on)h Fv(\(\012)p Fq(;)17 b Fr(F)10 b Fv(\))p Fg(.)512 4078 y Fh(\(c\))80 b Fq(\015)786 4041 y Fm(f)859 4078 y Fr(2)28 b Fv(\005)p Fg(.)313 4280 y Fh(2\))94 b(\(a\))81 b Fg(F)-8 b(or)21 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Fg(restricted)g(to)f Fq(b)1815 4881 y Fl(\000)p Fn(1)1927 4917 y Fv(\(\005)2038 4932 y Fn(1)2077 4917 y Fv(\))h Fg(is)f(one-to-one.)257 5127 y Fh(Remarks)257 5268 y(4.13)48 b Fv(The)30 b(set)g(of)f(stationary)g(non-n)m(ull)f(LIS) h(satisfying)g(Hulse)g(condition)f(is)h(strictly)478 5388 y(included)j(in)g(\002)1058 5403 y Fn(1)1097 5388 y Fq(:)1828 5637 y Fv(12)p eop %%Page: 13 13 13 12 bop 257 573 a Fh(4.14)48 b Fv(Since)37 b(for)f(all)e Fq(k)k Fr(\025)d Fv(1)p Fq(;)e Fv(v)-5 b(ar)1498 588 y Fm(k)1540 573 y Fv(\()p Fq(g)t Fv(\))34 b Fr(\025)h Fq(\016)1856 588 y Fm(k)1899 573 y Fv(\()p Fq(g)t Fv(\),)i(\002)2166 588 y Fn(1)2242 573 y Fv(includes)f(the)h(set)h(of)e(stationary)478 693 y(non-n)m(ull)31 b(LIS)i(with)f(summable)f(v)-5 b(ariation.)257 894 y(Consider)33 b(no)m(w)h(the)f(map)1416 1111 y Fq(c)28 b Fv(:)f(\005)1613 1126 y Fn(2)1681 1111 y Fr(!)g Fv(\002)1884 1126 y Fn(2)1951 1111 y Fq(;)44 b(\015)33 b Fr(7!)27 b Fq(f)2292 1070 y Fm(\015)3246 1111 y Fv(\(4.15\))257 1329 y(de\014ned)34 b(b)m(y)1091 1449 y Fq(f)1150 1402 y Fm(\015)1139 1476 y Fn(\003)1210 1368 y Fj(\000)1256 1449 y Fq(A)28 b Fr(j)f Fq(!)1473 1464 y Fn(\003)1522 1473 y Fe(\000)1578 1368 y Fj(\001)1679 1449 y Fk(,)105 b Fv(lim)1812 1512 y Fm(k)r Fl(!)p Fn(+)p Fl(1)2063 1449 y Fq(\015)2114 1487 y Fn(\003)p Fl([)p Fn(\003)2259 1452 y Ff(\()p Fi(k)q Ff(\))2259 1508 y(+)2366 1449 y Fv(\()o Fq(A)28 b Fr(j)g Fq(!)t Fv(\))583 b(\(4.16\))257 1659 y(for)32 b(all)f(\003)c Fr(2)h(S)791 1674 y Fm(b)826 1659 y Fq(;)44 b(A)28 b Fr(2)g(F)1164 1674 y Fn(\003)1217 1659 y Fv(,)33 b(and)f Fq(!)f Fr(2)d Fv(\012)33 b(for)f(whic)m(h)h(the) g(limit)d(exists.)257 1860 y Fh(Theorem)37 b(4.17)h(\(sp)s (eci\014cation)e Fk( )h Fh(LIS\))313 1980 y(1\))81 b Fg(The)33 b(map)f Fq(c)g Fg(is)g(w)m(ell)g(de\014ned.)45 b(That)33 b(is,)f(for)g Fq(\015)h Fr(2)28 b Fv(\005)2424 1995 y Fn(2)507 2183 y Fh(\(a\))81 b Fg(the)34 b(limit)d(\(4.16\))i (exists)i(for)e(all)f Fv(\003)e Fr(2)h(S)2230 2198 y Fm(b)2264 2183 y Fq(;)47 b(A)30 b Fr(2)h(F)2610 2198 y Fn(\003)2662 2183 y Fq(;)47 b(!)2797 2198 y Fn(\003)2846 2207 y Fe(\000)2932 2183 y Fr(2)31 b Fv(\012)3099 2198 y Fn(\003)3148 2207 y Fe(\000)3239 2183 y Fg(and)j(is)730 2303 y(indep)s(enden)m(t)g(of)e Fq(!)1455 2318 y Fn(\003)1504 2327 y Ff(+)1558 2303 y Fg(.)499 2464 y Fh(\(b\))81 b Fq(f)789 2428 y Fm(\015)865 2464 y Fg(is)33 b(a)f(LIS)h(on)f Fv(\(\012)p Fq(;)17 b Fr(F)10 b Fv(\))p Fg(.)512 2625 y Fh(\(c\))80 b Fq(f)789 2589 y Fm(\015)861 2625 y Fr(2)28 b Fv(\002)1031 2640 y Fn(2)1070 2625 y Fg(.)313 2828 y Fh(2\))94 b(\(a\))81 b Fr(G)6 b Fv(\()p Fq(f)892 2792 y Fm(\015)936 2828 y Fv(\))34 b(=)h Fr(G)6 b Fv(\()p Fq(\015)f Fv(\))34 b(=)h Fr(f)o Fq(\026)1569 2792 y Fm(\015)1614 2828 y Fr(g)o Fg(,)j(where)g Fq(\026)2073 2792 y Fm(\015)2153 2828 y Fg(is)e(the)h(only)f(Gibbs)g(measure)h(con-)730 2948 y(sisten)m(t)c(with)f Fq(\015)5 b Fg(.)499 3109 y Fh(\(b\))81 b Fg(The)33 b(map)f Fq(c)h Fg(is)f(one-to-one.)257 3295 y 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Fm(!)2655 5198 y(j)2709 5173 y Fv(\()p Fq(f)2795 5188 y Fm(i)2823 5173 y Fv(\))p 2086 5227 776 4 v 2087 5318 a(1)f(+)h Fq(c)2297 5284 y Fm(!)2297 5343 y(j)2348 5318 y Fv(\()p Fq(f)2434 5333 y Fm(i)2462 5318 y Fv(\))2500 5289 y Fl(\000)p Fn(1)2611 5318 y Fq(\016)2658 5284 y Fm(!)2654 5343 y(j)2708 5318 y Fv(\()p Fq(f)2794 5333 y Fm(i)2822 5318 y Fv(\))2871 5249 y Fq(:)1828 5637 y Fv(13)p eop %%Page: 14 14 14 13 bop 313 648 a Fh(2\))81 b Fg(F)-8 b(or)31 b Fq(\015)i Fr(2)28 b Fv(\005)919 663 y Fn(2)959 648 y Fg(,)k Fv(\003)c Fr(2)g(S)1268 663 y Fm(b)1335 648 y Fg(and)33 b Fq(j)g(<)28 b(l)1731 663 y Fn(\003)1784 648 y Fg(,)33 b Fq(\016)1891 607 y Fm(!)1887 672 y(j)1941 648 y Fv(\()p Fq(f)2038 601 y Fm(\015)2027 675 y Fn(\003)2082 648 y Fv(\))28 b Fr(\024)g Fv(1)22 b Fr(\000)2433 523 y Fn(+)p Fl(1)2432 553 y Fj(Y)2423 765 y Fm(i)p Fn(=)p Fm(l)2523 776 y Ff(\003)2595 571 y Fv(1)g Fr(\000)h Fq(c)2808 535 y Fm(!)2808 596 y(j)2858 571 y Fv(\()p Fq(\015)2947 586 y Fm(k)2990 571 y Fv(\))3028 535 y Fl(\000)p Fn(1)3138 571 y Fq(\016)3185 535 y Fm(!)3181 596 y(j)3236 571 y Fv(\()p Fq(\015)3325 586 y Fm(i)3353 571 y Fv(\))p 2595 625 796 4 v 2596 716 a(1)f(+)g Fq(c)2807 682 y Fm(!)2807 742 y(j)2858 716 y Fv(\()p Fq(\015)2947 731 y Fm(k)2989 716 y Fv(\))3027 687 y Fl(\000)p Fn(1)3138 716 y Fq(\016)3185 682 y Fm(!)3181 742 y(j)3235 716 y Fv(\()p Fq(\015)3324 731 y Fm(i)3352 716 y Fv(\))3400 648 y Fq(:)257 947 y Fv(Under)34 b(suitable)d (conditions)h(the)h(maps)f Fq(b)h Fv(and)g Fq(c)f Fv(are)h(recipro)s (cal.)257 1149 y Fh(Theorem)k(4.20)h(\(LIS)f Fk(!)h Fh(sp)s (eci\014cation\))270 1269 y(\(a\))82 b Fq(b)22 b Fr(\016)g Fq(c)28 b Fv(=)f(Id)98 b Fg(o)m(v)m(er)34 b Fq(c)1240 1233 y Fl(\000)p Fn(1)1334 1269 y Fv(\(\002)1448 1284 y Fn(1)1487 1269 y Fv(\))f Fg(and)g Fr(G)6 b Fv(\()p Fq(f)1910 1233 y Fm(\015)1954 1269 y Fv(\))27 b(=)h Fr(G)6 b Fv(\()p Fq(\015)f Fv(\))28 b(=)f Fr(f)p Fq(\026)2560 1233 y Fm(\015)2604 1269 y Fr(g)p Fg(.)262 1482 y Fh(\(b\))82 b Fq(c)22 b Fr(\016)g Fq(b)28 b Fv(=)f(Id)98 b Fg(o)m(v)m(er)34 b Fq(b)1239 1445 y Fl(\000)p Fn(1)1334 1482 y Fv(\(\005)1445 1497 y Fn(2)1484 1482 y Fv(\))f Fg(and)f Fr(G)6 b Fv(\()p Fq(\015)1903 1445 y Fm(f)1949 1482 y Fv(\))27 b(=)h Fr(G)6 b Fv(\()p Fq(f)11 b Fv(\))27 b(=)2449 1401 y Fj(\010)2507 1482 y Fq(\026)2566 1445 y Fm(f)2611 1401 y Fj(\011)2669 1482 y Fg(.)275 1685 y Fh(\(c\))81 b Fq(b)31 b Fg(and)g Fq(c)h Fg(establish)e(a)h(one-to-one)g(corresp)s(ondence)i(b)s(et)m(w)m (een)g Fv(\002)2914 1700 y Fn(3)2984 1685 y Fg(and)f Fv(\005)3246 1700 y Fn(3)3316 1685 y Fg(that)494 1805 y(preserv)m(es)j(the)e(consisten)m(t)h(measure.)404 1992 y Fv(W)-8 b(e)42 b(remark)f(that)g(\002)1219 2007 y Fn(3)1300 1992 y Fv(includes)h(the)g(w)m(ell)f(studied)h(pro)s(cesses)h(with)f (Hold)m(\177)-46 b(erian)257 2112 y(transition)41 b(rates)h(\(see,)j (for)d(instance,)i(Lalley)-8 b(,)43 b(1986,)h(or)d(Keller,)i(1998\).)71 b(P)m(art)42 b(\(c\))257 2233 y(of)34 b(the)g(theorem)f(sho)m(ws,)j(in) d(particular,)g(the)h(equiv)-5 b(alence)34 b(b)s(et)m(w)m(een)i(suc)m (h)f(pro)s(cesses)257 2353 y(and)e(Bo)m(w)m(en's)i(Gibbs)d(measures.) 257 2686 y Fw(5)161 b(Pro)t(ofs)257 2905 y Fv(W)-8 b(e)33 b(start)g(with)f(a)g(collection)f(of)h(results)h(used)h(for)e(sev)m (eral)h(pro)s(ofs.)257 3101 y Fh(Lemma)k(5.1)257 3221 y Fg(Consider)23 b Fv(\003)k Fr(2)h(S)7 b Fq(;)45 b Fv(\001)28 b Fr(\032)g Fv(\003)1264 3185 y Fm(c)1320 3221 y Fg(and)22 b Fq(\031)k Fg(a)21 b(probabilit)m(y)f(k)m(ernel)i(o)m(v)m(er)h Fv(\()p Fr(F)2722 3236 y Fn(\003)2797 3221 y Fr(\012)g(F)2969 3236 y Fn(\001)3048 3221 y Fq(;)33 b Fv(\012)3178 3236 y Fn(\003)3254 3221 y Fr(\002)22 b Fv(\012)3423 3236 y Fn(\001)3487 3221 y Fv(\))257 3341 y Fg(suc)m(h)34 b(that)f Fq(\031)20 b Fv(\()p Fq(A)875 3356 y Fn(\001)966 3341 y Fr(j)27 b(\001)17 b Fv(\))27 b(=)44 b Fx(1)-30 b Fv(1)1315 3356 y Fm(A)1368 3367 y Ff(\001)1426 3341 y Fv(\()17 b Fr(\001)g Fv(\))p Fq(;)43 b Fr(8)17 b Fq(A)1779 3356 y Fn(\001)1870 3341 y Fr(2)28 b(F)2036 3356 y Fn(\001)2098 3341 y 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Fv(\))55 b Fr(\024)h Fq(\031)21 b Fv(\()p Fq(A)2155 5295 y Fn(\003)2208 5280 y Fv(\))33 b Fx(1)-30 b Fv(1)2343 5295 y Fm(A)2396 5272 y Fi(c)2396 5318 y Ff(\001)2481 5280 y Fq(:)1828 5637 y Fv(14)p eop %%Page: 15 15 15 14 bop 257 573 a Fv(On)33 b(the)g(other)g(hand,)615 786 y Fq(\031)20 b Fv(\()p Fq(A)801 801 y Fn(\003)877 786 y Fr(\002)i Fq(A)1049 801 y Fn(\001)1112 786 y Fv(\))g(+)g Fq(\031)f Fv(\()p Fq(A)1457 801 y Fn(\003)1532 786 y Fr(\002)i Fq(A)1705 745 y Fm(c)1705 811 y Fn(\001)1768 786 y Fv(\))55 b(=)g Fq(\031)21 b Fv(\()p Fq(A)2179 801 y Fn(\003)2232 786 y Fv(\))33 b Fx(1)-30 b Fv(1)2367 801 y Fm(A)2420 812 y Ff(\001)2500 786 y Fv(+)22 b Fq(\031)e Fv(\()p Fq(A)2784 801 y Fn(\003)2837 786 y Fv(\))34 b Fx(1)-31 b Fv(1)2972 801 y Fm(A)3025 778 y Fi(c)3025 824 y Ff(\001)3111 786 y Fq(:)257 1000 y Fv(The)34 b(last)e(three)h (displa)m(ys)g(imply)847 1213 y Fq(\031)t Fv(\()p Fq(A)p Fv(\))27 b(=)h Fq(\031)20 b Fv(\()p Fq(A)1372 1228 y Fn(\003)1425 1213 y Fv(\))61 b Fx(1)-30 b 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y(l)2210 5165 y Ff(\003)2287 5128 y Fr(j)h Fq(!)2404 5143 y Fn(\003)2453 5152 y Fe(\000)2509 5047 y Fj(\001)p 1588 5181 1224 4 v 1588 5203 a(X)1617 5415 y Fm(\014)1657 5426 y Ff(\003)1748 5298 y Fq(f)1796 5313 y Fn([)p Fm(l)1837 5324 y Ff(\003)1883 5313 y Fm(;n)p Fn(])1985 5217 y Fj(\000)2031 5298 y Fq(\014)2086 5313 y Fn(\003)2156 5298 y Fq(!)2217 5313 y Fn(\003)2266 5294 y Fi(c)2298 5313 y Fl(\\)p Fn([)p Fm(l)2386 5324 y Ff(\003)2431 5313 y Fm(;n)p Fn(])2545 5298 y Fr(j)f Fq(!)2661 5313 y Fn(\003)2710 5322 y Fe(\000)2766 5217 y Fj(\001)1828 5637 y Fv(15)p eop %%Page: 16 16 16 15 bop 257 573 a Fv(and)33 b(the)g(factorization)1084 844 y Fq(f)1132 859 y Fn([)p Fm(k)r(;n)p Fn(])1293 763 y Fj(\000)1339 844 y Fq(!)1404 803 y Fm(n)1400 869 y(k)1478 844 y Fr(j)27 b Fq(!)1598 803 y Fm(k)r Fl(\000)p Fn(1)1594 869 y Fl(\0001)1730 763 y Fj(\001)1803 844 y Fv(=)1949 719 y Fm(n)1907 749 y Fj(Y)1912 961 y Fm(i)p Fn(=)p Fm(k)2051 844 y Fq(f)2099 859 y Fm(i)2144 763 y Fj(\000)2189 844 y Fq(!)2250 859 y Fm(i)2306 844 y Fr(j)g Fq(!)2426 803 y Fm(i)p Fl(\000)p Fn(1)2422 869 y Fl(\0001)2552 763 y Fj(\001)2642 844 y Fq(:)257 1154 y Fv(W)-8 b(e)48 b(then)g(apply)e (inequalities)g(\(5.3\))g(to)h(b)s(ound)g(eac)m(h)h(of)f(the)g(factors) g(b)m(y)h(similar)257 1274 y(factors)33 b(with)f(conditioning)e (con\014guration)i Fq(\033)t Fv(.)404 1420 y(T)-8 b(o)30 b(obtain)f Fh(\(5.5\))h Fv(w)m(e)h(apply)f(the)h(LIS-reconstruction)f (form)m(ula)f(\(3.4\))g(with)h Fq(m)e Fv(=)257 1540 y Fq(n)23 b Fv(+)f(1)32 b(whic)m(h)h(yields)257 1828 y Fq(F)320 1843 y Fn(\003)p Fm(;n)p Fn(+1)526 1828 y Fv(\()p Fq(!)625 1843 y Fn(\003)706 1828 y Fr(j)27 b Fq(!)t Fv(\))g(=)1427 1752 y Fq(f)1475 1768 y Fn([)p Fm(l)1516 1779 y Ff(\003)1562 1768 y Fm(;n)p Fn(])1664 1671 y Fj(\000)1710 1752 y Fq(!)1775 1716 y Fm(n)1771 1778 y(l)1792 1789 y Ff(\003)1869 1752 y Fr(j)h Fq(!)1986 1767 y Fn(\003)2035 1776 y Fe(\000)2091 1671 y Fj(\001)2169 1752 y Fq(f)2217 1767 y Fm(n)p Fn(+1)2371 1671 y Fj(\000)2417 1752 y Fq(!)2478 1767 y Fm(n)p Fn(+1)2642 1752 y Fr(j)g Fq(!)2759 1768 y Fn(])p Fl(\0001)p Fm(;n)p Fn(])2990 1671 y Fj(\001)p 1004 1805 2454 4 v 1004 1867 a(X)1034 2079 y Fm(\014)1074 2090 y Ff(\003)1165 1821 y Fj(\024)1218 1961 y Fq(f)1266 1977 y Fn([)p Fm(l)1307 1988 y Ff(\003)1352 1977 y Fm(;n)p Fn(])1455 1881 y Fj(\000)1500 1961 y Fq(\014)1555 1976 y Fn(\003)1625 1961 y Fq(!)1686 1977 y Fn(\003)1735 1958 y Fi(c)1767 1977 y Fl(\\)p Fn([)p Fm(l)1855 1988 y Ff(\003)1900 1977 y Fm(;n)p Fn(])2014 1961 y Fr(j)f Fq(!)2130 1976 y Fn(\003)2179 1985 y Fe(\000)2235 1881 y Fj(\001)2314 1961 y Fq(f)2362 1976 y Fm(n)p Fn(+1)2516 1881 y Fj(\000)2562 1961 y Fq(!)2623 1976 y Fm(n)p Fn(+1)2787 1961 y Fr(j)g Fq(\014)2897 1976 y Fn(\003)2967 1961 y Fq(!)3028 1977 y Fn(])p Fl(\0001)p Fm(;n)p Fn(])p Fl(n)p Fn(\003)3343 1881 y Fj(\001)3406 1821 y(\025)3496 1828 y Fq(:)257 2294 y Fv(In)36 b(the)f(denominator,)f(only)g Fq(\014)1422 2309 y Fn(\003)1510 2294 y Fv(with)g Fq(f)1782 2309 y Fm(n)p Fn(+1)1936 2214 y Fj(\000)1982 2294 y Fq(!)2043 2309 y Fm(n)p Fn(+1)2207 2294 y Fr(j)28 b Fq(!)2324 2310 y Fn(])p Fl(\0001)p Fm(;n)p Fn(])p Fl(n)p Fn(\003)2656 2294 y Fq(\014)2711 2309 y Fn(\003)2764 2214 y Fj(\001)2841 2294 y Fr(6)p Fv(=)j(0)k(con)m(tribute.)257 2415 y(W)-8 b(e)33 b(use)h(inequalities)d(\(5.3\))h(for)g(these.)44 b Fk(\003)257 2616 y Fh(Lemma)37 b(5.6)257 2736 y Fg(Let)29 b Fq(f)38 b Fr(2)28 b Fv(\002)684 2751 y Fn(1)723 2736 y Fq(;)45 b(!)31 b Fr(2)d Fv(\012)g Fg(and)h Fv(\003)e Fr(2)h(S)7 b Fg(.)43 b(Consider)28 b(the)h(sequence)i Fq(F)2620 2751 y Fn(\003)p Fm(;n)2752 2736 y Fv(\()p Fq(!)2851 2751 y Fn(\003)2931 2736 y Fr(j)d Fq(!)t Fv(\))15 b Fq(;)34 b(n)28 b Fr(\025)g Fq(m)3442 2751 y Fn(\003)257 2857 y Fg(de\014ned)34 b(b)m(y)g(\(4.10\).)43 b(Then)33 b(the)g(limit)c(\(4.11\))j(exits)h(and)g(satis\014es)727 2990 y Fj(\014)727 3049 y(\014)727 3109 y(\014)760 3104 y Fq(\015)816 3057 y Fm(f)811 3131 y Fn(\003)880 3104 y Fv(\()p Fq(!)979 3119 y Fn(\003)1060 3104 y Fr(j)27 b Fq(!)1176 3119 y Fn(\003)1225 3100 y Fi(c)1261 3104 y Fv(\))22 b Fr(\000)h Fq(F)1484 3119 y Fn(\003)p Fm(;n)1616 3104 y Fv(\()p Fq(!)1715 3119 y Fn(\003)1795 3104 y Fr(j)28 b Fq(!)t Fv(\))1953 2990 y Fj(\014)1953 3049 y(\014)1953 3109 y(\014)2014 3104 y Fr(\024)2160 3010 y Fj(X)2119 3222 y Fm(k)r Fl(\025)p Fm(n)p Fn(+1)2362 3104 y Fq(c)2404 3063 y Fm(!)2404 3129 y Fn(\003)2457 3104 y Fv(\()p Fq(f)2543 3119 y Fm(k)2586 3104 y Fv(\))2624 3063 y Fl(\000)p Fn(1)2735 3104 y Fq(\016)2782 3063 y Fm(!)2778 3129 y Fn(\003)2832 3104 y Fv(\()p Fq(f)2918 3119 y Fm(k)2961 3104 y Fv(\))p Fq(:)269 b Fv(\(5.7\))257 3426 y Fh(Pro)s(of)108 b Fv(F)-8 b(rom)31 b(\(5.5\),)h(plus)h(the)g(fact)f(that)g Fq(F)1994 3441 y Fn(\003)p Fm(;n)2138 3426 y Fr(2)c Fv([0)p Fq(;)17 b Fv(1])27 b Fr(8)17 b Fq(n)28 b Fr(\025)g Fq(m;)33 b Fv(w)m(e)g(obtain)718 3646 y Fr(j)o Fq(F)808 3661 y Fn(\003)p Fm(;n)p Fn(+1)1031 3646 y Fv(\()p Fq(!)1130 3661 y Fn(\003)1210 3646 y Fr(j)28 b Fq(!)t Fv(\))21 b Fr(\000)i Fq(F)1553 3661 y Fn(\003)p Fm(;n)1685 3646 y 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Fr(j)k Fq(!)t Fv(\))p Fr(j)o Fv(.)40 b(In)22 b(particular)3092 3905 y Fj(\000)3138 3986 y Fq(F)3201 4001 y Fn(\003)p Fm(;n)3317 3986 y Fv(\()p Fq(!)3416 4001 y Fn(\003)3496 3986 y Fr(j)257 4122 y Fq(!)t Fv(\))360 4042 y Fj(\001)405 4161 y Fm(n)p Fl(\025)p Fm(m)592 4122 y Fv(is)h(a)f(Cauc)m(h)m(y)j(sequence)g(so)e(the)g(limit) d(lim)2113 4137 y Fm(n)p Fl(!)p Fn(+)p Fl(1)2373 4122 y Fq(F)2436 4137 y Fn(\003)p Fm(;n)2568 4122 y Fv(\()p Fq(!)2667 4137 y Fn(\003)2747 4122 y Fr(j)28 b Fq(!)t Fv(\))f Fk(,)g Fq(\015)3094 4075 y Fm(f)3089 4149 y Fn(\003)3158 4122 y Fv(\()p Fq(!)3257 4137 y Fn(\003)3338 4122 y Fr(j)g Fq(!)t Fv(\))257 4256 y(exits)33 b(and)g(satis\014es)g(\(5.7\))f(for)g (eac)m(h)i Fq(!)d Fr(2)d Fv(\012)p Fq(:)98 b Fk(\003)257 4545 y Fd(5.1)136 b(LIS)44 b Fc( )h Fd(sp)t(eci\014cation)257 4729 y Fh(Pro)s(of)38 b(of)f(Theorem)g(4.12)404 4875 y Fv(Lemma)31 b(\(5.6\))h(pro)m(v)m(es)i(Item)f Fh(1\))k(\(a\))p Fv(.)404 5020 y(T)-8 b(o)26 b(pro)m(v)m(e)h(item)d Fh(1\))29 b(\(b\))p Fv(,)e(w)m(e)g(observ)m(e)g(that)f Fq(\015)2083 4973 y Fm(f)2078 5047 y Fn(\003)2130 5020 y Fv(\()p Fq(A)i Fr(j)g(\001)17 b Fv(\))24 b(is)i(clearly)f Fr(F)2901 5035 y Fn(\003)2950 5016 y Fi(c)2985 5020 y Fv(-measurable)257 5140 y(for)43 b(ev)m(ery)i(\003)g Fr(2)h(S)k Fv(and)44 b(ev)m(ery)g Fq(A)i Fr(2)g(F)10 b Fv(.)74 b(Moreo)m(v)m(er)45 b(condition)c(\(b\))i(of)g(De\014nition)257 5261 y(2.12)36 b(together)h(with)f(the)i(presence)g(of)e(the)i(indicator)c(function)53 b Fx(1)-30 b Fv(1)2819 5276 y Fm(!)2863 5290 y Ff(\003)2904 5276 y Fi(c)2936 5290 y Fe(\\)p Ff([)p Fi(l)3016 5307 y Ff(\003)3061 5290 y Fi(;m)3134 5307 y Ff(\003)3179 5290 y(])3242 5261 y Fv(in)36 b(the)1828 5637 y(16)p eop %%Page: 17 17 17 16 bop 257 573 a Fv(denominator)37 b(of)h(\(4.10\))g(imply)e(that)i Fq(\015)1789 526 y Fm(f)1784 600 y Fn(\003)1837 573 y Fv(\()p Fq(B)k Fr(j)37 b(\001)17 b Fv(\))37 b(=)54 b Fx(1)-30 b Fv(1)2370 588 y Fm(B)2430 573 y Fv(\()17 b Fr(\001)g Fv(\))37 b(for)h(ev)m(ery)i(\003)d Fr(2)h(S)46 b Fv(and)257 693 y(ev)m(ery)35 b Fq(B)e Fr(2)28 b(F)788 708 y Fn(\003)837 689 y Fi(c)872 693 y Fv(.)44 b(Therefore)34 b(it)d(su\016ces)k(to)d(sho)m(w)i(that)812 814 y Fj(X)798 1023 y Fm(!)842 1037 y Ff(\001)p Fe(n)p Ff(\003)986 908 y Fq(\015)1042 861 y Fm(f)1037 935 y Fn(\003)1106 908 y Fv(\()p Fq(!)1205 923 y Fn(\003)1285 908 y Fr(j)28 b Fq(!)1402 923 y Fn(\003)1451 904 y Fi(c)1487 908 y Fv(\))16 b Fq(\015)1597 861 y Fm(f)1592 935 y Fn(\001)1671 828 y Fj(\000)1717 908 y Fq(!)1778 924 y Fn(\001)p Fl(n)p Fn(\003)1953 908 y Fr(j)27 b Fq(!)2069 923 y Fn(\001)2128 904 y Fi(c)2164 828 y Fj(\001)2265 908 y Fv(=)55 b Fq(\015)2452 861 y Fm(f)2447 935 y Fn(\001)2526 908 y Fv(\()p Fq(!)2625 923 y Fn(\003)2706 908 y Fr(j)27 b Fq(!)2822 923 y Fn(\001)2881 904 y Fi(c)2917 908 y Fv(\))340 b(\(5.8\))257 1220 y(for)32 b(eac)m(h)h(\003)p Fq(;)17 b Fv(\001)27 b Fr(2)i(S)40 b Fv(suc)m(h)33 b(that)f(\003)27 b Fr(\032)i Fv(\001)j(and)g(eac)m(h)h Fq(!)e Fr(2)e Fv(\012.)43 b(Let)33 b(us)f(denote,)i(for)d(eac)m(h)257 1341 y(\000)d Fr(\022)g Fv(\001,)33 b(eac)m(h)h(in)m(teger)e Fq(n)c Fr(\025)g Fq(l)1354 1356 y Fn(\000)1435 1341 y Fv(and)33 b Fq(!)1690 1304 y Fm(n)1686 1365 y Fl(\0001)1843 1341 y Fr(2)28 b Fv(\012)2007 1304 y Fm(n)2007 1365 y Fl(\0001)2137 1341 y Fv(,)973 1578 y Fq(G)1050 1593 y Fn(\000)p Fm(;n)1177 1578 y Fv(\()17 b Fr(\001)27 b(j)g Fq(!)1403 1593 y Fn(\000)1447 1574 y Fi(c)1483 1578 y Fv(\))55 b Fk(,)g Fq(f)1757 1593 y Fn([)p Fm(l)1798 1604 y Ff(\000)1840 1593 y Fm(;n)p Fn(])1926 1467 y Fj(\020)2002 1578 y Fr(\001)33 b Fx(1)-30 b Fv(1)2127 1593 y Fm(!)2171 1607 y Ff(\000)2209 1593 y Fi(c)2240 1607 y Fe(\\)p Ff([)p Fi(l)2320 1624 y Ff(\000)2360 1607 y Fi(;n)p Ff(])2471 1463 y Fj(\014)2471 1523 y(\014)2471 1583 y(\014)2532 1578 y Fq(!)2593 1593 y Fn(\000)2637 1602 y Fe(\000)2693 1467 y Fj(\021)2753 1578 y Fq(:)257 1808 y Fv(De\014nition)31 b(\(4.10\){\(4.11\))g(b)s(ecomes)1069 2067 y Fq(\015)1120 2082 y Fn(\001)1199 2067 y Fv(\()p Fq(h)d Fr(j)g Fq(!)t Fv(\))54 b(=)107 b(lim)1665 2126 y Fm(n)p Fl(!)p Fn(+)p Fl(1)1931 1999 y Fq(G)2008 2014 y Fn(\001)p Fm(;n)2150 1999 y Fv(\()16 b Fx(1)-30 b Fv(1)2268 2014 y Fm(!)2312 2025 y Ff(\001)2397 1999 y Fr(j)28 b Fq(!)2514 2014 y Fn(\001)2573 1995 y Fi(c)2608 1999 y Fv(\))p 1931 2044 716 4 v 1997 2135 a Fq(G)2074 2150 y Fn(\001)p Fm(;n)2216 2135 y Fv(\(1)g Fr(j)f Fq(!)2447 2150 y Fn(\001)2506 2131 y Fi(c)2542 2135 y Fv(\))2656 2067 y Fq(:)612 b Fv(\(5.9\))257 2329 y(Using)35 b(the)h(reconstruction)g(prop)s(ert)m(y) g(\(3.4\))e(of)h(LIS)h(with)e Fq(l)h Fv(=)d Fq(l)2709 2344 y Fn(\001)2772 2329 y Fq(;)49 b(n)32 b Fv(=)h Fq(l)3076 2344 y Fn(\003)3153 2329 y Fr(\000)24 b Fv(1)35 b(and)257 2449 y Fq(m)28 b Fv(=)g Fq(n)p Fv(,)33 b(w)m(e)g(obtain)620 2652 y Fq(G)697 2667 y Fn(\001)p Fm(;n)822 2652 y Fv(\()17 b Fx(1)-31 b Fv(1)940 2667 y Fm(!)984 2681 y Ff(\001)p Fe(n)p Ff(\003)1143 2652 y Fr(j)28 b Fq(!)1260 2667 y Fn(\001)1319 2648 y Fi(c)1354 2652 y Fv(\))56 b(=)f Fq(G)1656 2667 y Fn(\001)p 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b(Let)34 b(us)h(\014x)g Fq(k)2224 4179 y Fn(0)2298 4164 y Fv(suc)m(h)g(that)f Fq(\017)2771 4179 y Fm(k)2845 4164 y Fq(<)c Fv(1)k(for)g Fq(k)g Fr(\025)d Fq(k)3429 4179 y Fn(0)3468 4164 y Fv(.)257 4284 y(By)i(\(4.6\))f(and)h(the)g(lo)m(w)m(er)g(b)s(ound)g(in)e (\(5.5\))1336 4563 y Fq(\015)1392 4516 y Fm(f)1387 4590 y Fn(\003)1495 4563 y Fr(\025)56 b Fq(F)1691 4578 y Fn(\003)p Fm(;k)1797 4587 y Ff(0)1916 4439 y Fl(1)1887 4468 y Fj(Y)1868 4681 y Fm(k)r Fn(=)p Fm(k)1999 4690 y Ff(0)2033 4563 y Fv(\(1)22 b Fr(\000)h Fq(\017)2281 4578 y Fm(k)2324 4563 y Fv(\))k Fq(:)257 4860 y Fv(The)33 b(righ)m(t-hand)d(side)i(is)e (strictly)h(p)s(ositiv)m(e)g(on)g(\012)h(due)g(to)f(the)h(non-n)m (ullness)f(of)g Fq(F)3351 4875 y Fn(\003)p Fm(;k)3457 4884 y Ff(0)257 4990 y Fv(and)i(the)g(summabilit)m(y)d(of)i(the)h Fq(\017)1492 5005 y Fm(k)1535 4990 y Fv(.)43 b(Hence)34 b Fq(\015)1951 4942 y Fm(f)1946 5017 y Fn(\003)2032 4990 y Fv(is)e(non-n)m(ull)f(on)h(\012.)404 5133 y(T)-8 b(o)32 b(pro)m(v)m(e)i(assertion)f Fh(2\)\(a\))e Fv(w)m(e)j(consider)f Fq(\026)27 b Fr(2)h(G)2277 5052 y Fj(\000)2323 5133 y Fq(f)2382 5096 y Fn(\()p Fm(V)t(;!)r Fn(\))2552 5052 y Fj(\001)2630 5133 y Fv(and)k(denote)801 5373 y Fq(G)878 5322 y Fn(\()p Fm(V)t(;!)r Fn(\))878 5400 y(\003)p Fm(;n)1065 5373 y Fv(\()17 b Fr(\001)27 b(j)g Fq(\033)1285 5388 y Fn(\003)1334 5369 y Fi(c)1371 5373 y Fv(\))55 b Fk(,)g Fq(f)1645 5389 y Fn([)p Fm(l)1686 5400 y Ff(\003)1731 5389 y Fm(;n)p Fn(])1817 5263 y Fj(\020)1893 5373 y Fr(\001)33 b Fx(1)-30 b Fv(1)2018 5388 y Fm(\033)2058 5402 y Ff(\003)2099 5388 y Fi(c)2131 5402 y Fe(\\)p Ff([)p Fi(l)2211 5419 y Ff(\003)2256 5402 y Fi(;n)p Ff(])2367 5259 y Fj(\014)2367 5318 y(\014)2367 5378 y(\014)2428 5373 y Fq(!)2489 5388 y Fm(V)2530 5397 y Fe(\000)2602 5373 y Fq(\033)2657 5389 y Fn(\003)2706 5398 y Fe(\000)2759 5389 y Fl(n)p Fm(V)2835 5398 y Fe(\000)2892 5263 y Fj(\021)1828 5637 y Fv(17)p eop %%Page: 18 18 18 17 bop 257 573 a Fv(for)37 b(all)e(\003)g Fr(2)h(S)816 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2052 y(Finally)36 b(w)m(e)j(pro)m(v)m(e)h Fh(2\))k(\(c\))p Fv(.)61 b(Let)38 b Fq(f)1769 2016 y Fn(1)1847 2052 y Fv(and)h Fq(f)2102 2016 y Fn(2)2179 2052 y Fv(b)s(e)g(t)m(w)m(o)h(LIS)e(on)h(\(\012)p Fq(;)17 b Fr(F)10 b Fv(\),)40 b(b)s(oth)e(in)257 2172 y Fq(b)298 2136 y Fl(\000)p Fn(1)393 2172 y Fv(\(\005)504 2187 y Fn(1)544 2172 y Fv(\),)25 b(and)g(suc)m(h)g(that)f Fq(\015)1286 2136 y Fm(f)1327 2113 y Ff(1)1394 2172 y Fv(=)j Fq(\015)1553 2136 y Fm(f)1594 2113 y Ff(2)1633 2172 y Fq(:)d Fv(By)h(2\))e(\(c\),)j Fq(\026)2170 2136 y Fm(f)2211 2113 y Ff(1)2278 2172 y Fv(=)h Fq(\026)2440 2136 y Fm(f)2481 2113 y Ff(2)2548 2172 y Fk(,)g Fq(\026)p Fv(.)40 b(The)25 b(non-n)m(ullness)257 2293 y(of)36 b Fq(f)431 2257 y Fn(1)507 2293 y Fv(and)h Fq(f)760 2257 y Fn(2)835 2293 y Fv(on)g(\012)g(implies)d(that)i Fq(\026)g Fv(c)m(harges)i(all)c(op)s(en)j(sets)h(in)e(\012.)55 b(Therefore,)39 b Fq(f)3453 2257 y Fn(1)3442 2318 y(\003)257 2413 y Fv(and)24 b Fq(f)497 2377 y Fn(2)486 2438 y(\003)563 2413 y 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Fn(+)p Fl(1)1839 5185 y Fv(Spr)2002 5104 y Fj(\000)2048 5185 y Fq(f)2096 5200 y Fn([)p Fm(i)p Fl(\000)p Fm(n;i)p Fn(])2305 5185 y Fq(h)2361 5104 y Fj(\001)2434 5185 y Fv(=)c(0)f Fq(;)605 b Fv(\(5.13\))730 5388 y Fg(then)33 b Fr(jG)6 b Fv(\()p Fq(f)11 b Fv(\))p Fr(j)27 b(\024)h Fv(1)p Fg(.)1828 5637 y Fv(18)p eop %%Page: 19 19 19 18 bop 499 573 a Fh(\(b\))81 b Fg(If)40 b Fq(f)50 b Fg(is)40 b(con)m(tin)m(uous)h(and)f Fr(j)o(G)6 b Fv(\()p Fq(f)11 b Fv(\))p Fr(j)40 b(\024)h Fv(1)p Fg(,)h(then)e(\(5.13\))f(is)h (v)m(eri\014ed)h(for)e(all)730 693 y Fq(i)28 b Fr(2)g Fo(Z)i Fg(and)i(all)f(con)m(tin)m(uous)i Fr(F)1869 708 y Fl(\024)p Fm(i)1952 693 y Fg(-measurable)e(con)m(tin)m(uous)j (function)e Fq(h)p Fg(.)257 880 y Fh(Pro)s(of)108 b Fv(W)-8 b(e)40 b(pro)s(of)f(part)h(1\),)i(the)e(pro)s(of)f(of)h(2\))g(is)f (similar.)63 b(The)41 b(ob)m(vious)f(spread-)257 1000 y(reducing)33 b(relation)1127 1220 y(inf)1118 1285 y Fb(e)-43 b Fm(!)s Fl(2)p Fn(\012)1275 1220 y Fq(h)p Fv(\()t Fj(e)-59 b Fq(!)t Fv(\))55 b Fr(\024)h Fv(\()p Fq(\015)1749 1235 y Fn(\003)1801 1220 y Fq(h)p Fv(\)\()p Fq(!)t Fv(\))f Fr(\024)h Fv(sup)2229 1304 y Fb(e)-43 b Fm(!)r Fl(2)p Fn(\012)2387 1220 y Fq(h)p Fv(\()t Fj(e)-59 b Fq(!)t Fv(\))27 b Fq(;)257 1496 y Fv(v)-5 b(alid)41 b(for)h(ev)m(ery)i(b)s (ounded)f(measurable)f(function)g Fq(h)h Fv(on)f(\012)h(and)f(ev)m(ery) j(con\014gura-)257 1616 y(tion)e Fq(!)49 b Fr(2)e Fv(\012,)f(plus)d (the)h(consistency)h(condition)d(\(2.4\))h(imply)e(that)j(the)f (sequence)257 1736 y Fr(f)p Fv(sup)q(\()p Fq(\015)543 1751 y Fn(\003)592 1759 y Fi(n)638 1736 y Fq(h)p Fv(\))p Fr(g)32 b Fv(is)f(decreasing)h(\(and)f(b)s(ounded)h(b)s(elo)m(w)g(b)m (y)g(inf)23 b Fq(h)p Fv(\),)32 b(while)e(the)i(sequence)257 1857 y Fr(f)p Fv(inf)6 b(\()p Fq(\015)514 1872 y Fn(\003)563 1880 y Fi(n)609 1857 y Fq(h)p Fv(\))p Fr(g)31 b Fv(is)f(increasing)f (\(and)i(b)s(ounded)g(ab)s(o)m(v)m(e)g(b)m(y)h(sup)17 b Fq(h)p Fv(\).)43 b(Therefore,)32 b(if)d Fq(\026;)17 b(\027)34 b Fr(2)257 1977 y(G)6 b Fv(\()p 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Fn([)p Fm(n;p)p Fn(])2936 1516 y Fv(\()p Fq(A)28 b Fr(j)f Fq(\033)t Fv(\))h Fq(:)257 1741 y Fv(In)33 b(a)g(similar)c(w)m(a)m(y)34 b(w)m(e)f(obtain)1165 1962 y Fq(\015)1216 1977 y Fn([)p Fm(n;p)p Fn(+)p Fm(k)r Fn(])1451 1962 y Fv(\()p Fq(A)28 b Fr(j)f Fq(\021)t Fv(\))55 b Fr(\024)1953 1894 y Fv(1)p 1933 1939 91 4 v 1933 2030 a Fq(K)2049 1962 y(\015)2100 1977 y Fn([)p Fm(n;p)p Fn(])2241 1962 y Fv(\()p Fq(A)28 b Fr(j)f Fq(\033)t Fv(\))h Fq(:)257 2169 y Fv(W)-8 b(e)33 b(conclude)g(that)g(for)f(eac)m(h)h Fq(k)e Fr(\025)d Fv(0)1115 2351 y Fq(\015)1166 2366 y Fn([)p Fm(n;p)p Fn(+)p Fm(k)r Fn(])1400 2351 y Fv(\()p Fq(A)g Fr(j)f Fq(\030)5 b Fv(\))55 b Fr(\025)h Fq(K)1958 2310 y Fn(2)2014 2351 y Fq(\015)2065 2366 y Fn([)p Fm(n;p)p Fn(+)p Fm(k)r Fn(])2299 2351 y Fv(\()p Fq(A)28 b Fr(j)f Fq(\021)t Fv(\))h Fq(:)257 2533 y Fv(Letting)44 b Fq(k)52 b Fr(!)c(1)d Fv(w)m(e)h(obtain,)h(due)e(to)g(de\014nition)f(\(4.16\),)j(that)e Fq(f)2891 2485 y Fm(\015)2880 2564 y Fn([)p Fm(n;m)p Fn(])3048 2533 y Fv(\()p Fq(A)j Fr(j)h Fq(\030)5 b Fv(\))48 b Fr(\025)257 2668 y Fq(K)347 2632 y Fn(2)387 2668 y Fq(f)446 2621 y Fm(\015)435 2699 y Fn([)p Fm(n;m)p Fn(])603 2668 y Fv(\()p Fq(A)28 b Fr(j)f Fq(\021)t Fv(\).)141 b Fk(\003)257 2951 y Fd(5.3)136 b(LIS)44 b Fc(!)h Fd(sp)t (eci\014cation)257 3136 y Fh(Pro)s(of)38 b(of)f(Theorem)g(4.19)257 3256 y Fv(Assertion)42 b Fh(1\))47 b(\(a\))41 b Fv(is)g(a)h(direct)f (consequence)j(of)d(inequalit)m(y)g(\(5.7\))g(of)g(Lemma)f(5.6)257 3377 y(with)33 b(\003)27 b(=)g Fr(f)p Fq(k)s Fr(g)33 b Fv(and)f Fq(n)c Fv(=)g Fq(j)g Fr(\000)23 b Fv(1.)404 3516 y(Assertion)35 b Fh(1\))k(\(b\))34 b Fv(follo)m(ws)f(from)h(the)h Fq(n)c Fr(!)g(1)j Fv(limit)d(of)j(inequalities)f(\(5.4\))h(and)257 3637 y(the)f(fact)g(that)f(0)c Fr(\024)g Fq(F)1074 3652 y Fn(\003)p Fm(;n)1217 3637 y Fr(\024)g Fv(1.)404 3777 y(T)-8 b(o)31 b(pro)m(v)m(e)i(assertion)f Fh(2\))p Fv(,)f(let)g Fq(k)s(;)17 b(j)33 b Fr(2)28 b Fo(Z)h Fv(suc)m(h)k(that)f Fq(j)h(<)28 b(k)34 b Fv(and)e(consider)g 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y(By)49 b(the)f(sp)s(eci\014cation)f(reconstruction)h(form)m(ula)e (\(3.9\))h(with)g(\003)53 b(=)g Fr(f)p Fq(n)33 b Fv(+)f(1)p Fr(g)47 b Fv(and)257 4894 y(\000)28 b(=)g([)p Fq(l)506 4909 y Fn(\003)559 4894 y Fq(;)17 b(n)p Fv(])32 b(w)m(e)i(ha)m(v)m(e) 470 5147 y Fq(\015)521 5162 y Fn([)p Fm(l)562 5173 y Ff(\003)607 5162 y Fm(;n)p Fn(+1])800 5066 y Fj(\000)845 5147 y Fq(\033)900 5162 y Fn(\003)982 5147 y Fr(j)27 b Fq(\033)1092 5162 y Fn(\003)1141 5171 y Fe(\000)1214 5147 y Fq(\033)1273 5106 y Fn(+)p Fl(1)1269 5171 y Fm(m)1331 5182 y Ff(\003)1378 5171 y Fn(+1)1472 5066 y Fj(\001)1573 5147 y Fv(=)1743 5052 y Fj(X)1704 5263 y Fm(\033)1746 5240 y Fi(n)1744 5284 y(m)1798 5301 y Ff(\003)1844 5284 y(+1)2116 5073 y Fq(\015)2167 5088 y Fm(n)p Fn(+1)2321 4992 y Fj(\000)2366 5073 y Fq(\033)2421 5088 y Fm(n)p Fn(+1)2586 5073 y Fr(j)h Fq(\033)2701 5032 y Fm(n)2697 5098 y Fl(\0001)2843 5073 y Fq(\033)2902 5032 y Fn(+)p Fl(1)2898 5098 y Fm(n)p Fn(+2)3036 4992 y Fj(\001)p 1952 5124 1294 4 v 1955 5200 a(X)1952 5412 y Fm(\030)1983 5421 y Fi(n)p Ff(+1)2206 5221 y Fq(\015)2257 5236 y Fm(n)p Fn(+1)2411 5140 y Fj(\000)2456 5221 y Fq(\030)2499 5236 y Fm(n)p Fn(+1)2664 5221 y Fr(j)f Fq(\033)2778 5184 y Fm(n)2774 5245 y Fl(\0001)2921 5221 y Fq(\033)2980 5179 y Fn(+)p Fl(1)2976 5245 y Fm(n)p Fn(+2)3113 5140 y Fj(\001)p 2129 5271 1107 4 v 2129 5368 a Fq(\015)2180 5384 y Fn([)p Fm(l)2221 5395 y Ff(\003)2266 5384 y Fm(;n)p Fn(])2369 5287 y Fj(\000)2415 5368 y Fq(\033)2474 5334 y Fm(n)2470 5396 y(l)2491 5407 y Ff(\003)2569 5368 y Fr(j)g Fq(\033)2679 5383 y Fn(\003)2728 5392 y Fe(\000)2801 5368 y Fq(\030)2844 5383 y Fm(n)p Fn(+1)2997 5368 y Fq(\033)3056 5327 y Fn(+)p Fl(1)3052 5392 y Fm(n)p Fn(+2)3190 5287 y Fj(\001)3256 5147 y Fq(:)1828 5637 y Fv(22)p eop %%Page: 23 23 23 22 bop 257 573 a Fv(Using)39 b(\(5.22\))f(and)h(\(5.23\))f(it)g(is)h (easy)h(to)f(sho)m(w,)i(b)m(y)f(induction)e(o)m(v)m(er)i Fq(n)f Fr(\025)g Fq(m)3237 588 y Fn(\003)3317 573 y Fv(+)26 b(1,)257 693 y(that)349 964 y Fq(\015)400 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b(1.33)g(in)e(Georgii)g(\(1988\).)67 b(In)41 b(the)g(latter,)h(the)f(order-consistency)502 814 y(condition)25 b(\(A.9\))g(is)h(replaced)g(b)m(y)g(the)g (requiremen)m(t)h(that)e(the)h(singletons)f(come)502 934 y(from)31 b(a)g(pre-existing)g(sp)s(eci\014cation)g(\(whic)m(h)h (the)g(prescription)f(reconstructs\).)257 1137 y Fh(A.11)47 b Fv(Iden)m(tit)m(y)33 b(\(ii\))c(can)i(b)s(e)h(used,)g(in)f(fact,)g (to)g(inductiv)m(ely)g(de\014ne)h(the)g(family)d Fq(\032)i Fv(b)m(y)502 1258 y(adding)41 b(one)g(site)g(at)f(a)h(time.)67 b(In)42 b(fact,)h(this)e(is)f(what)h(is)g(done)g(in)g(the)g(pro)s(of) 502 1378 y(b)s(elo)m(w.)55 b(The)37 b(inequalities)d(\(iv\))i(relate)f (the)i(non-n)m(ullness)f(prop)s(erties)g(of)g Fq(\032)h Fv(to)502 1499 y(those)d(of)e(the)h(original)c(family)h Fr(f)p Fq(\032)1787 1514 y Fm(i)1815 1499 y Fr(g)1865 1518 y Fm(i)p Fl(2)p Fp(Z)1987 1499 y Fi(d)r Fv(.)257 1702 y Fh(A.12)47 b Fv(In)35 b(the)g(case)h Fq(E)41 b Fv(coun)m(table,)35 b Fq(\025)1676 1717 y Fm(i)1704 1702 y Fv(=coun)m(ting)f(measure,)i(the)f(order-consistency)502 1822 y(requiremen)m(t)29 b(\(A.9\))g(is)f(automatically)d(v)m (eri\014ed)30 b(if)d(the)i(singletons)f(are)h(de\014ned)502 1943 y(through)k(a)f(measure)h Fq(\026)f Fv(on)h Fr(F)42 b Fv(in)32 b(the)h(form)1479 2212 y Fq(\032)1529 2227 y Fm(i)1558 2212 y Fv(\()p Fq(!)t Fv(\))55 b(=)79 b(lim)1885 2272 y Fm(n)p Fl(!1)2160 2145 y Fq(\026)p Fv(\()p Fq(!)2318 2160 y Fm(V)2359 2168 y Fi(n)2405 2145 y Fv(\))p 2095 2189 413 4 v 2095 2280 a Fq(\026)p Fv(\()p Fq(!)2253 2296 y Fm(V)2294 2304 y Fi(n)2336 2296 y Fl(nf)p Fm(i)p Fl(g)2470 2280 y Fv(\))502 2495 y(for)33 b(an)h(exhausting)f(sequence)j (of)d(v)m(olumes)h Fr(f)p Fq(V)2280 2510 y Fm(n)2326 2495 y Fr(g)p Fv(.)46 b(Indeed,)35 b(a)e(simple)f(compu-)502 2616 y(tation)g(sho)m(ws)i(that)e(the)h(last)f(t)m(w)m(o)h(terms)g(in)f (\(A.9\))g(coincide)g(with)1653 2891 y(lim)1629 2950 y Fm(n)p Fl(!1)1930 2823 y Fq(\026)p Fv(\()p Fq(!)2088 2838 y Fm(V)2129 2846 y Fi(n)2175 2823 y Fv(\))p 1839 2868 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588 y Fm(i)2789 573 y Fq(\025)2846 588 y Fm(i)2874 573 y Fv(\))28 b(=)f Fq(\026;)44 b Fr(8)17 b Fq(i)28 b Fr(2)g Fv(\003,)494 693 y(then)33 b Fq(\026)17 b Fv(\()o Fq(\032)879 708 y Fn(\003)949 693 y Fq(\025)1006 708 y Fn(\003)1059 693 y Fv(\))28 b(=)f Fq(\026:)249 897 y Fh(\(I5\))81 b Fv(\()p Fq(\032)604 912 y Fn(\003)p Fl([)p Fn(\000)765 897 y Fq(\025)822 912 y Fn(\003)p Fl([)p Fn(\000)966 897 y Fv(\))17 b(\()p Fq(\032)1109 912 y Fm(i)1154 897 y Fq(\025)1211 912 y Fm(i)1239 897 y Fv(\))27 b(=)h Fq(\032)1458 912 y Fn(\003)p Fl([)p Fn(\000)1619 897 y Fq(\025)1676 912 y Fn(\003)p Fl([)p Fn(\000)1820 897 y Fq(;)45 b Fr(8)17 b Fq(i)27 b Fr(2)i Fv(\003)21 b Fr([)i Fv(\000)p Fq(:)257 1100 y Fv(Let)43 b(us)g(\014rst)g(commen)m(t)f(wh)m(y)i(these)g(prop)s(erties)e(imply)f (the)i(theorem.)73 b(It)43 b(is)f(clear)257 1220 y(that)d(prop)s (erties)f(\(I3\){\(I5\),)i(together)f(with)f(the)i(deterministic)d(c)m (haracter)i(of)f(\()p Fq(\025)3404 1235 y Fn(\003)3457 1220 y Fv(\))257 1341 y(on)45 b 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Fj(\021)257 5388 y Fv(whic)m(h)h(is)g(precisely)f Fq(\032)1083 5403 y Fn(\003)p Fl([)p Fn(\000)1260 5388 y Fv(according)g(to)h(our)f (de\014nition)g(\(A.13\).)1828 5637 y(28)p eop %%Page: 29 29 29 28 bop 404 573 a Fh(\(I3\))40 b Fv(W)-8 b(e)42 b(assume)g(that)f Fr(j)p Fv(\003)p Fr(j)i(\025)g Fv(2,)h(otherwise)e(\(I3\))f(is)g(just)h (the)g(normalization)257 693 y(h)m(yp)s(othesis)34 b(\(A.6\).)43 b(Let)33 b Fq(k)e Fv(=)c(max)17 b(\003.)43 b(De\014nition)31 b(A.13)h(and)h(prop)s(ert)m(y)g(A.3)f(yield)684 1000 y Fq(\025)741 1015 y Fn(\003)811 1000 y Fv(\()p Fq(\032)899 1015 y Fn(\003)952 1000 y Fv(\))55 b(=)h Fq(\025)1234 1015 y Fm(k)1293 830 y Fj( )1462 923 y Fq(\025)1519 938 y Fn(\003)1568 915 y Fe(\003)1568 962 y Fi(k)1627 842 y Fj(\000)1673 923 y Fq(\032)1723 938 y Fn(\003)1772 915 y Fe(\003)1772 962 y Fi(k)1814 842 y Fj(\001)p 1382 978 559 4 v 1382 1074 a Fq(\025)1439 1089 y Fn(\003)1488 1066 y Fe(\003)1488 1113 y Fi(k)1546 993 y Fj(\000)1592 1074 y Fq(\032)1642 1089 y Fn(\003)1691 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Fq(\032)2035 1591 y Fm(k)2094 1576 y Fq(\032)2144 1535 y Fl(\000)p Fn(1)2144 1603 y(\003)2193 1581 y Fe(\003)2193 1627 y Fi(k)2239 1465 y Fj(\021)2332 1576 y Fq(\032)2382 1591 y Fn(\003)2431 1568 y Fe(\003)2431 1614 y Fi(k)2490 1576 y Fq(\032)2540 1535 y Fl(\000)p Fn(1)2540 1604 y Fm(k)2662 1576 y Fq(;)257 1772 y Fv(therefore)1230 1998 y Fq(\025)1287 2013 y Fn(\003)1357 1998 y Fv(\()p Fq(\032)1445 2013 y Fn(\003)1498 1998 y Fv(\))h(=)1733 1895 y Fq(\025)1790 1910 y Fm(k)1849 1784 y Fj(\020)1908 1895 y Fq(\032)1958 1910 y Fm(k)2018 1895 y Fq(\032)2068 1854 y Fl(\000)p Fn(1)2068 1923 y(\003)2117 1900 y Fe(\003)2117 1946 y Fi(k)2163 1784 y Fj(\021)p 1733 1976 490 4 v 1733 2102 a Fq(\025)1790 2117 y Fm(k)1849 1991 y Fj(\020)1908 2102 y Fq(\032)1958 2117 y Fm(k)2018 2102 y Fq(\032)2068 2061 y Fl(\000)p Fn(1)2068 2130 y(\003)2117 2107 y Fe(\003)2117 2153 y Fi(k)2163 1991 y Fj(\021)2287 1998 y Fv(=)g(1)27 b Fq(:)404 2374 y Fh(\(I4\))i Fv(T)-8 b(o)31 b(a)m(v)m(oid)f(a)g (trivialit)m(y)e(w)m(e)k(assume)f(that)f Fr(j)p Fv(\003)p Fr(j)d(\025)h Fv(2.)42 b(Let)31 b Fq(\026)f Fv(b)s(e)h(a)f(probabilit)m (y)257 2495 y(measure)j(on)f(\(\012)p Fq(;)17 b Fr(F)10 b Fv(\))32 b(suc)m(h)i(that)e Fq(\026)17 b Fv(\()o Fq(\032)1670 2510 y Fm(i)1715 2495 y Fq(\025)1772 2510 y Fm(i)1800 2495 y Fv(\))28 b(=)f Fq(\026)32 b Fv(for)g(all)e Fq(i)e Fr(2)g Fv(\003.)43 b(Consider)33 b Fq(k)e Fv(=)c(max)17 b(\003)257 2615 y(and)33 b(a)e(measurable)h(function)f Fq(h)p Fv(.)44 b(By)32 b(the)h(factorization)d(prop)s(ert)m(y)j (\(A.3\))e(of)h Fq(\025)3253 2630 y Fn(\003)3338 2615 y Fv(and)257 2736 y(the)h(de\014nition)f(\(A.13\))g(of)g Fq(\032)1326 2751 y Fn(\003)1380 2736 y Fv(,)g(w)m(e)i(ha)m(v)m(e)714 3073 y Fq(\026)773 2962 y Fj(\020)832 3073 y Fv(\()p Fq(\032)920 3088 y Fn(\003)973 3073 y Fq(\025)1030 3088 y Fn(\003)1083 3073 y Fv(\))17 b(\()p Fq(h)p Fv(\))1270 2962 y Fj(\021)1385 3073 y Fv(=)55 b Fq(\026)1592 2872 y Fj(2)1592 3052 y(4)1658 3073 y Fq(\025)1715 3088 y Fn(\003)1764 3065 y Fe(\003)1764 3111 y Fi(k)1822 2872 y Fj(0)1822 3052 y(@)1910 2932 y(\022)2191 3005 y Fq(\032)2241 3020 y Fm(k)p 1993 3050 V 1993 3176 a Fq(\025)2050 3191 y Fm(k)2109 3066 y Fj(\020)2169 3176 y Fq(\032)2219 3191 y Fm(k)2278 3176 y Fq(\032)2328 3135 y Fl(\000)p Fn(1)2328 3204 y(\003)2377 3181 y Fe(\003)2377 3228 y Fi(k)2423 3066 y Fj(\021)2509 3073 y Fq(\025)2566 3088 y Fm(k)2608 2932 y Fj(\023)2682 3073 y Fv(\()p Fq(h)p Fv(\))2814 2872 y Fj(1)2814 3052 y(A)2901 2872 y(3)2901 3052 y(5)3012 3073 y Fq(:)257 3424 y Fv(By)34 b(the)f(inductiv)m(e)g(h)m(yp)s (othesis)h(\(I4\))e Fq(\026)h Fv(is)f(consisten)m(t)i(with)e Fq(\032)2591 3439 y Fn(\003)2640 3416 y Fe(\003)2640 3462 y Fi(k)2683 3424 y Fq(\025)2740 3439 y Fn(\003)2789 3416 y Fe(\003)2789 3462 y Fi(k)2863 3424 y Fv(and)h(with)g Fq(\032)3326 3439 y Fm(k)3369 3424 y Fq(\025)3426 3439 y Fm(k)3468 3424 y Fv(,)257 3544 y(th)m(us)551 3856 y Fq(\026)610 3746 y Fj(\020)686 3856 y Fv(\()p Fq(\032)774 3871 y Fn(\003)827 3856 y Fq(\025)884 3871 y Fn(\003)937 3856 y Fv(\))17 b(\()p Fq(h)p Fv(\))1124 3746 y Fj(\021)1238 3856 y Fv(=)56 b Fq(\026)1446 3656 y Fj(2)1446 3835 y(4)1511 3856 y Fv(\()p Fq(\032)1599 3871 y Fm(k)1659 3856 y Fq(\025)1716 3871 y Fm(k)1758 3856 y Fv(\))1813 3656 y Fj(0)1813 3835 y(@)1900 3856 y Fq(\032)1950 3815 y Fl(\000)p Fn(1)1950 3884 y(\003)1999 3861 y Fe(\003)1999 3908 y Fi(k)2073 3716 y Fj(\022)2354 3789 y Fq(\032)2404 3804 y Fm(k)p 2156 3833 V 2156 3960 a Fq(\025)2213 3975 y Fm(k)2272 3849 y Fj(\020)2331 3960 y Fq(\032)2381 3975 y Fm(k)2441 3960 y Fq(\032)2491 3919 y Fl(\000)p Fn(1)2491 3987 y(\003)2540 3965 y Fe(\003)2540 4011 y Fi(k)2586 3849 y Fj(\021)2672 3856 y Fq(\025)2729 3871 y Fm(k)2771 3716 y Fj(\023)2845 3856 y Fv(\()p Fq(h)p Fv(\))2977 3656 y Fj(1)2977 3835 y(A)3064 3656 y(3)3064 3835 y(5)3175 3856 y Fq(:)257 4202 y Fv(But,)39 b(in)d(the)i(righ)m(t-hand)e(side,)i(the)g(t)m(w)m(o) g(innermost)e(in)m(tegrals)g(with)h(resp)s(ect)h(to)f Fq(\025)3453 4217 y Fm(k)257 4322 y Fv(comm)m(ute)32 b(with)h(the)g(external)f(one,)h(so)g(w)m(e)h(ha)m(v)m(e)549 4654 y Fq(\026)608 4543 y Fj(\020)667 4654 y Fv(\()p Fq(\032)755 4669 y Fn(\003)809 4654 y Fq(\025)866 4669 y Fn(\003)919 4654 y Fv(\))16 b(\()p Fq(h)p Fv(\))1105 4543 y Fj(\021)1248 4654 y Fv(=)83 b Fq(\026)1483 4453 y Fj(2)1483 4633 y(4)1548 4654 y Fv(\()p Fq(\032)1636 4669 y Fm(k)1696 4654 y Fq(\025)1753 4669 y Fm(k)1795 4654 y Fv(\))1850 4453 y Fj(0)1850 4633 y(@)2164 4586 y Fq(h)p 1947 4631 V 1947 4757 a(\025)2004 4772 y Fm(k)2063 4647 y Fj(\020)2123 4757 y Fq(\032)2173 4772 y Fm(k)2232 4757 y Fq(\032)2282 4716 y Fl(\000)p Fn(1)2282 4785 y(\003)2331 4762 y Fe(\003)2331 4809 y Fi(k)2377 4647 y Fj(\021)2544 4654 y Fq(\025)2601 4669 y Fm(k)2660 4543 y Fj(\020)2720 4654 y Fq(\032)2770 4669 y Fm(k)2829 4654 y Fq(\032)2879 4612 y Fl(\000)p Fn(1)2879 4681 y(\003)2928 4658 y Fe(\003)2928 4705 y Fi(k)2974 4543 y Fj(\021)3033 4453 y(1)3033 4633 y(A)3121 4453 y(3)3121 4633 y(5)1248 4970 y Fv(=)g Fq(\026)1466 4859 y Fj(\020)1525 4970 y Fv(\()p Fq(\032)1613 4985 y Fm(k)1672 4970 y Fq(\025)1729 4985 y Fm(k)1772 4970 y Fv(\))16 b(\()p Fq(h)p Fv(\))1958 4859 y Fj(\021)2046 4970 y Fq(;)257 5217 y Fv(whic)m(h)33 b(pro)m(v)m(es)i(\(I4\).)1828 5637 y(29)p eop %%Page: 30 30 30 29 bop 404 573 a Fh(\(I5\))24 b Fv(Denote)h(\001)j(=)g(\003)7 b Fr([)g Fv(\000)26 b(and)g(pic)m(k)g Fq(i;)17 b(j)33 b Fr(2)28 b Fv(\001,)f Fq(i)h Fr(6)p Fv(=)g Fq(j)j Fv(and)26 b(a)f(measurable)g(function)257 693 y Fq(h)p Fv(.)44 b(By)33 b(\(A.3\))f(and)h(\(I2\))f(w)m(e)i(ha)m(v)m(e)688 934 y Fj(\020)747 1044 y Fq(\032)797 1059 y Fn(\001)877 1044 y Fq(\025)934 1059 y Fn(\001)997 934 y Fj(\021\020)1116 1044 y Fv(\()p Fq(\032)1204 1059 y Fm(i)1249 1044 y Fq(\025)1306 1059 y Fm(i)1334 1044 y Fv(\))16 b(\()p Fq(h)p Fv(\))1520 934 y Fj(\021)1635 1044 y Fv(=)56 b Fq(\025)1824 1059 y Fm(j)1877 844 y Fj(2)1877 1023 y(4)1953 830 y(\020)2012 941 y Fq(\032)2062 956 y Fn(\001)2121 933 y Fe(\003)2121 978 y Fi(j)2178 941 y Fq(\025)2235 956 y Fn(\001)2294 933 y Fe(\003)2294 978 y Fi(j)2334 830 y Fj(\021)q(\020)2453 941 y Fv(\()p Fq(\032)2541 956 y Fm(i)2586 941 y Fq(\025)2643 956 y Fm(i)2671 941 y Fv(\))17 b(\()p Fq(h)p Fv(\))2858 830 y Fj(\021)p 1953 1021 965 4 v 2134 1148 a Fq(\025)2191 1163 y Fn(\001)2250 1140 y Fe(\003)2250 1185 y Fi(j)2307 1037 y Fj(\020)2366 1148 y Fq(\032)2416 1163 y Fn(\001)2475 1140 y Fe(\003)2475 1185 y Fi(j)2532 1148 y Fq(\032)2582 1107 y Fl(\000)p Fn(1)2582 1173 y Fm(j)2677 1037 y Fj(\021)2927 844 y(3)2927 1023 y(5)3038 1044 y Fq(:)257 1395 y Fv(Therefore,)34 b(applying)d(inductiv)m(e)i(assumption)f(\(I5\))g(w)m(e)i(obtain)868 1636 y Fj(\020)928 1746 y Fq(\032)978 1761 y Fn(\001)1058 1746 y Fq(\025)1115 1761 y Fn(\001)1178 1636 y Fj(\021\020)1297 1746 y Fv(\()p Fq(\032)1385 1761 y Fm(i)1430 1746 y Fq(\025)1487 1761 y Fm(i)1515 1746 y Fv(\))16 b(\()p Fq(h)p Fv(\))1701 1636 y Fj(\021)1844 1746 y Fv(=)83 b Fq(\025)2060 1761 y Fm(j)2113 1546 y Fj(2)2113 1725 y(4)2196 1532 y(\020)2255 1643 y Fq(\032)2305 1658 y Fn(\001)2364 1635 y Fe(\003)2364 1680 y Fi(j)2421 1643 y Fq(\025)2478 1658 y Fn(\001)2537 1635 y Fe(\003)2537 1680 y Fi(j)2577 1532 y Fj(\021)2653 1643 y Fv(\()p Fq(h)p Fv(\))p 2189 1723 602 4 v 2189 1850 a Fq(\025)2246 1865 y Fn(\001)2305 1842 y Fe(\003)2305 1887 y Fi(j)2362 1739 y Fj(\020)2421 1850 y Fq(\032)2471 1865 y Fn(\001)2530 1842 y Fe(\003)2530 1887 y Fi(j)2587 1850 y Fq(\032)2637 1809 y Fl(\000)p Fn(1)2637 1875 y Fm(j)2732 1739 y Fj(\021)2801 1546 y(3)2801 1725 y(5)1844 2064 y Fv(=)g(\()p Fq(\032)2091 2079 y Fn(\001)2171 2064 y Fq(\025)2228 2079 y Fn(\001)2290 2064 y Fv(\))17 b(\()p Fq(h)p Fv(\))28 b Fq(:)97 b Fk(\003)516 b Fv(\(A.21\))257 2397 y Fw(References)257 2616 y Fv([Ber87])111 b(H.)46 b(Berb)s(ee.)85 b(Chains)47 b(with)f(complete)f(connections:)72 b(Uniqueness)48 b(and)670 2736 y(Mark)m(o)m(v)34 b(represen)m(tation.) 44 b Fs(Pr)-5 b(ob.)35 b(Th.)f(R)-5 b(el.)34 b(Fields)p Fv(,)e(76:243{53,)f(1987.)257 2940 y([BF)m(G9b])49 b(X.)37 b(Bressaud,)j(R.)c(F)-8 b(ern\023)-49 b(andez,)40 b(and)d(A.)g(Galv)m (es.)56 b(Deca)m(y)38 b(of)f(correlations)670 3060 y(for)c(non)g(H\177) -49 b(olderian)32 b(dynamics.)i(a)f(coupling)f(approac)m(h.)46 b Fs(Ele)-5 b(ct.)36 b(J.)f(Pr)-5 b(ob.)p Fv(,)670 3180 y(4,)32 b(1999b.)43 b(\()p Fa(http://www.math.washington)q(.ed)q(u/)n Fj(e)t Fa(ejpecp/)p Fv(\).)257 3384 y([Bo)m(w75])76 b(R.)28 b(Bo)m(w)m(en.)38 b Fs(Equilibrium)30 b(states)h(and)g(the)g(er)-5 b(go)g(dic)30 b(the)-5 b(ory)31 b(of)f(Anosov)h(dif-)670 3504 y(fe)-5 b(omorphisms)p Fv(.)41 b(Springer-V)-8 b(erlag,)30 b(Berlin-Heideb)s(erg-New)h(Y)-8 b(ork,)33 b(1975.)257 3708 y([Dob68])83 b(R.)39 b(L.)g(Dobrushin.)63 b(The)40 b(description)e(of)h(a)g(random)f(\014eld)h(b)m(y)h(means)g(of)670 3828 y(conditional)27 b(probabilities)f(and)k(conditions)e(of)h(its)g (regularit)m(y)-8 b(.)37 b Fs(The)-5 b(ory)32 b(of)670 3948 y(pr)-5 b(ob)g(ability)34 b(and)h(its)g(applic)-5 b(ations)p Fv(,)31 b(13:197{224,)g(1968.)257 4152 y([FM03])108 b(R.)31 b(F)-8 b(ern\023)-49 b(andez)32 b(and)g(G.)f(Maillard.)38 b(Chains)32 b(with)f(complete)f(connections.)670 4272 y(General)f(theory)-8 b(,)31 b(uniqueness,)h(loss)e(of)f(memory)g(and)h (mixing)e(prop)s(erties.)670 4393 y Fs(In)34 b(pr)-5 b(ep)g(ar)g(ation)p Fv(,)32 b(2003.)257 4596 y([FP97])131 b(R.)47 b(F)-8 b(ern\023)-49 b(andez)47 b(and)g(C.-E.)h(P\014ster.)87 b(Global)45 b(sp)s(eci\014cations)i(and)g(non-)670 4716 y(quasilo)s(calit)m(y)22 b(of)i(pro)5 b(jections)25 b(of)e(Gibbs)h (measures.)31 b Fs(The)c(A)n(nnals)f(of)i(Pr)-5 b(ob-)670 4837 y(ability)p Fv(,)33 b(25\(3\):1284{1315,)c(1997.)257 5040 y([Geo74])92 b(H.-O.)39 b(Georgii.)60 b(Sto)s(c)m(hastisc)m(he)40 b(Felder)e(und)i(ihre)e(An)m(w)m(endung)k(auf)c(In-)670 5161 y(teraktionssysteme.)45 b Fs(L)-5 b(e)g(ctur)g(e)35 b(Notes,)g(Institut)g(f)q(\177)-51 b(ur)36 b(A)n(ngewandte)d(Mathe-)670 5281 y(matik,)h(Universit\177)-50 b(at)35 b(Heidelb)-5 b(er)g(g)p Fv(,)32 b(1974.)1828 5637 y(30)p eop %%Page: 31 31 31 30 bop 257 573 a Fv([Geo88])92 b(H.-O.)36 b(Georgii.)51 b Fs(Gibbs)38 b(Me)-5 b(asur)g(es)38 b(and)g(Phase)f(T)-7 b(r)i(ansitions)p Fv(.)52 b(W)-8 b(alter)36 b(de)670 693 y(Gruyter)e(\(de)f(Gruyter)h(Studies)g(in)e(Mathematics,)h(V)-8 b(ol.)33 b(9\),)g(Berlin{New)670 814 y(Y)-8 b(ork,)33 b(1988.)257 1017 y([Har55])101 b(T.)24 b(E.)h(Harris.)j(On)c(c)m(hains) h(of)e(in\014nite)g(order.)29 b Fs(Paci\014c)e(J.)g(Math.)p Fv(,)f(5:707{24,)670 1137 y(1955.)257 1341 y([Hul97])106 b(P)-8 b(.)42 b(Hulse.)70 b(A)42 b(class)f(of)g(unique)h Fq(g)t Fv(-measure.)69 b Fs(Er)-5 b(go)g(d.)43 b(Th.)g(and)f(Dynam.)670 1461 y(Sys.)p Fv(,)32 b(17:1383{1392,)f(1997.)257 1665 y([Kal90])108 b(S.)39 b(Kalik)m(o)m(w.)59 b(Random)38 b(Mark)m(o)m(v)i(pro)s(cesses)g(and)f(uniform)d(martingales.)670 1785 y Fs(Isr.)e(J.)h(Math.)p Fv(,)e(71\(1\):33{54,)e(1990.)257 1988 y([Kea72])93 b(M.)25 b(Keane.)30 b(Strongly)24 b(mixing)e Fq(g)t Fv(-measures.)30 b Fs(Inventiones)25 b(Math.)p Fv(,)h(16:309{)670 2109 y(24,)32 b(1972.)257 2312 y([Kel98])114 b(G.)34 b(Keller.)48 b Fs(Equilibrium)36 b(States)h(in)f(Er)-5 b(go)g(dic)37 b(The)-5 b(ory)p Fv(.)48 b(Cam)m(bridge)34 b(Uni-)670 2433 y(v)m(ersit)m(y)g(Press,)g(1998.)257 2636 y([Lal86])123 b(S.)23 b(P)-8 b(.)24 b(Lalley)-8 b(.)26 b(Regeneration)d(represen)m(tation)h(for)e(one-dimensional)f (Gibbs)670 2756 y(states.)44 b Fs(A)n(nn.)34 b(Pr)-5 b(ob.)p Fv(,)33 b(14:1262{71,)d(1986.)257 2960 y([Lan73])97 b(O.)24 b(E.)g(Lanford.)29 b Fs(Entr)-5 b(opy)27 b(and)g(e)-5 b(quilibrium)26 b(states)h(in)g(classic)-5 b(al)25 b(statistic)-5 b(al)670 3080 y(me)g(chanics)p Fv(.)65 b(In:)60 b(A.)41 b(Lenard)g(\(ed.\),)i(Statiscal)c(mec)m(hanics)i(and)f(mathe-)670 3200 y(matical)30 b(problems,)j(Battelle)e(Seattle)i(Rencon)m(tres)h (1971,)f(LPNh)g(20,)g(pp.)670 3321 y(1-113.,)e(1973.)257 3524 y([Led74])103 b(F.)30 b(Ledrappier.)41 b(Princip)s(e)30 b(v)-5 b(ariationnel)28 b(et)j(syst)m(\022)-46 b(emes)33 b(dynamiques)e(sym-)670 3645 y(b)s(oliques.)39 b Fs(Z.)32 b(Wahrscheinlichkeitsthe)-5 b(orie)30 b(verw.)i(Gebiete)p Fv(,)f(30:185{202,)670 3765 y(1974.)257 3968 y([OM35])96 b(O.)45 b(Onicescu)g(and)g(G.)f(Miho)s(c.)80 b(Sur)45 b(les)f(c)m(ha)-11 b(^)-38 b(\020nes)46 b(statistiques.)80 b Fs(C.)45 b(R.)670 4089 y(A)-5 b(c)g(ad.)35 b(Sci.)f(Paris)p Fv(,)e(200:511|12,)f(1935.)257 4292 y([Rue78])92 b(D.)26 b(Ruelle.)32 b Fs(Thermo)-5 b(dynamic)27 b(formalism)p Fv(.)32 b(Encyclop)s(edia)26 b(of)g(Mathemat-)670 4413 y(ics)32 b(5,)h(New-Y)-8 b(ork:)44 b(Addison)32 b(W)-8 b(esley)g(,)34 b(1978.)257 4616 y([W)-8 b(al75])92 b(P)-8 b(.)49 b(W)-8 b(alters.)90 b(Ruelle's)47 b(op)s(erator)h(theorem)g(and) h Fq(g)t Fv(-measures.)89 b Fs(T)-7 b(r)i(ans.)670 4736 y(A)n(mer.)34 b(Math.)h(So)-5 b(c.)p Fv(,)32 b(214:375{87,)f(1975.)1828 5637 y(31)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0305010554172--