; TeX output 2002.08.26:1044썠:-iӍ-sandwithstheedualsequence(WnP)n0,?de nedinSection0,thisimpliesbrymonotonicity@s([0;t])p=](liminf"n!192)qīR P(X xڍn URt)(dx)p]liminfq n!1iP(X 0ڍn URt)p=]liminfq n!1iP(WnURt)p=]P(WURt)p]P(W<UR1);swhicrhfortUR!1yieldstheassertion..2.ˌTVopprorvetheconverse,plet(Y0;Z0)bSeindependenrtof(YnP;Zn);n92N;swithqdistribution.ThenZ0s+Y0WvisdistributedasW,henceUR=L(W)qisasstationary%distributionforP.Moreorver,4hthe%recurrenceofoisaconsequencesof@(2.2b),bbSecausethepotenrtialkernelGUR:=P㋟n0"hUPƟ2n satis es@bymonotonicity@sG(0;[0;t])}u=̟qīRuG(x;[0;t])(dx)~]==̟Pn0P(PƟ nJ)([0;t])~]==̟Pn0P([0;t])forall9ZtUR0: 8Tq lasy102.TVokderivrethestationarydistributionassoSciatedwithapositivrerecurrentsdistribution,(8.2)isnotofmruchuse.8FVorsomeexplicitexamples,obtainedsbryanadhoScprincipleintheframeworkofexpSonentialfamilies,see[7]..Thenextresultstrengthens(2.3):s(8.3)Prop`osition.is35pffositiverecurrentwheneverP(Y=UR0)>0.sPrffoof.SincexYn=UR0forsomen2Nwithprobabilitry1,theseriesin(8.2)isas nitesumalmostsurely: 2.InthisconrtextitisworthwhiletopSointoutthefollowing:s|TheJstationarydistributioncanbSecomputed,)atleastinprinciple,from@3&썠:-iӍ}-stheprobabilitiespUR:=P(Y=0)andqË:=UR1pandthedistributions0candǟ20sof(Y;Zܞ)undertheconditionY#9=0andY6=0,respSectivrely(theassumptionsq>6=E0 isnorealrestriction).TVothisendlet(Y0;Z0) and(Y2p0RAnP;Z2ܞ0RAn);nE2N;sbSexDindependenrtwithdistributions0 8Handǟ20,respectivrelyV,andxDdenotebryW20RAnstheNanalogueofWn {with(Y2p0RAm;Z2ܞ0RAm);mPznP);s(2)1UR=y0V>y1>:::uD!0and<,PI|en2HN_ynPzn<1;@5D1썠:-iӍ}-s(3)0UR=p0V1).IndepSendenrtlywofZnP;nUT2N;wletnowYnP;nUT2N;sbSeindependenrtwithadistributionysuchthat@sP(Y>URynP)=pnPfor.*n0:sThen0URURynPzn)ńrP(Y1;:::ʜ;Yn1>URynP)+P(Zn>znP)ŝ=rp n1ڍn/t+qnPfor.*nUR2N:sTherefore,brythesummabilityin(1)and(3),withprobability1@sY1:::Yn1ZnURynPznPforalmostalldn2N;shence,brythesummabilityin(2),theassertionfollowsfrom(8.2): 2.NorwPthecloserelationshipbSetweenthebSehaviourofananerecursions(XnP)n0andtheassoSciatedrandomwralk(Sn)n0canbSesummarized.>UndersaIwreakbSoundednessconditiononZܞ,isatis edforinstanceinthemultiplicativesmoSdel,thereisinfactabijection:s(8.7)Theorem.With}}arbitrffary h>UR0letoneofthefollowingtwoconditionssbffe35satis ed:s(a)@wkE(Zܞ 'jURYp) for35some?4 n<1;!Qns(b)AP(Y=UR0)=0andEf`f`ō hZ $Qmfe m  Y؟f`* #Sf`-<UR1:sThen35thefollowingtrichotomyholds:@sSn!UR+1=)trSansient;@sSn!UR1=)nulClvrSecurrent;@sSn!UR1=)positivn9e35rSecurrent:sPrffoof.1. ClearlyV,*[ xe@18marybSeassumedand =e@1willbSeconsidereds rst.́ThenqthecaseSn 5!:+1issettledbry(2.4),@whilethecaseSn!:1sfollorwsƑfrom(8.4).̚ThusitremainstoconsiderthecaseSn s!ˢ1.̚Undersconditiont(a)clearlyE(log-KT+IZܞ)UR<1tand(8.5a)applies,Kwhileunderconditions(b)similarlyE(log-KT+I(Z5=Yp))UR<1.8Inviewof@sW=URP㋟n2HN$Y1:::Ynō +Zn +Qmfe $  Yn stheassertionfollorwsfrom(8.2)asintheproSofof(8.5).@6S썠:-iӍ}-.2.FVorV h<UR1thetransienrtcaseisagainsettledby(2.4),bSecausethepassagesfromm(Yp20j;Zܞ20)UR=(Yp2 ;Zܞ2 5)mto(Y;Zܞ)doSesnota ecttheasymptoticbeharviourofstheassoSciatedrandomwralk.8Therecurrentcasefollowsfromtheinequalitiesߍ@sZ1Y2:::YnR+:::+ZnUR(Z ܞ0ڍ1Y p0ڍ2j:::5EY p0ڍn+:::+Z ܞ0ڍnP)Oǽ133x|feӻ !;Z @sZ1j+:::+Y1:::Yn1ZnUR(Z ܞ0ڍ1+:::+Y p0ڍ1j:::5EY p0ڍn1Z ܞ0ڍnP)Oǽ133x|feӻ !:~sIndeed,ifxfeR 20PandWƟ20ZdenotetheanaloguesofxfeRandW,thismeans@sx@sfeRJvUR(xfeRR 0 {)Oǽ133x|feӻ !andDMW(WƟ 0o)Oǽ133x|feӻ !: 2.It shouldbSemenrtionedthatconditions(a)and(b)accordingto(3.4)cansbSesimpli edto@sE(Zܞ 5)UR<1forsomeFy| h>0;sprorvidedYandZFareindepSendentorYisbSoundedawayfrom0..ThissectionwillbSeconcludedbryastabilityresult./ERelated,ӛbutnotcom-sparable,цconditionsYforconrtinuousYdepSendenceofone canbefoundin[5]s(seealso[6],crhapter9.1).8Thefollowingresultstrengthens(6.5):s(8.8)Prop`osition.LffetN43pk rwg !andk }bepositiverecurrentwithsta-stionary35distributionsk Vsuchthat@s n:=URsupk62HN+w˟qīR4"tyn9 cdkx<UR1and<Ȅ:=supk62HN+w˟qīRz ~xdkx<1sfor35some h>UR0.fiThens(a)@wkis35positivn9erSecurrent;~s(b)Ak Wwg x!8;swherffe35isthestationarydistributionassociatedwith.sPrffoof.(a)Inviewofthewreakconvergenceߍ@sqīRIyn9 cdUR and=HqīRF?z ~xd;swhicrhbythehypSotheseson and](andJensen'sinequality)yields@s1URqīR log-Byn7d<0and<,qīRGlogVT+]zd<1:sThrusoispSositiverecurrentaccordingto(8.5a)..(b)sPSincetheL#fcmti8p0{normsincreasewiththeorder,zclearly :1marybSesassumed.8Inthiscase@sqīRIx dkt*=g%qīRΟqRw(xy+z) k#(dx)k(dyn9;dz)tg%qīRΟqRw((xyn9) ?+z ~z)k#(dx)k(dyn9;dz)tg%(qīRx dk#) Q+;@7`썠:-iӍ}-swhicrhimpliesw/@ssupT$k62HNh`qīRqA x dkxōYQmfeI  1 $[<UR1:>ލsInVviewof h>UR0,"gtherefore,theVsequence(k#)k62HNb)isuniformlytighrtinM1(R+x)sandeacrhlimitpSointisexcessivewithrespSecttoДaccordingto(4.6),:becauseswreakconvergenceimpliesvXagueconvergence.C:Theassertionthusfollowsfroms(5.6): 2.Inthisresulttheconditionson a$and-:bSothcanbeshorwntobeessenrtial.sThenfollorwingcounterexamplesresemblethosein[5],represent,however,evensmoreextremesituations.8InbSothcasesthenotationPk :referstok#.s(1)Pk#(Z1=ōkQmfe Lq  yn9k))UR=ō N1Qmfe V(p ljz -9k?E;Pk(Z=URz)=1ō /1۟Qmfe V(p ljz -9kswithz5>UR0andY=yË2]0;1[yieldpSositivrerecurrentk#;ko2URN;suchthat.L@sk Wwg x!8"(yI{;zV)'@andO.k Wwg x!"1 :sThe rstconrvergenceisclearandthesecondonefollorwsfrom@sPk#(WURkg)0SPk#(Zi,=URzfor1ikg)I=S(1ō /1۟Qmfe V(p ljz -9k) kx!UR0:%is(2)Pk#(Y=UR0)=ō:O1Qmfe C  kg3G;Pk(Y=ōM1Qmfe  k ?C)=1ō/1۟Qmfe  k ;Pk(Y=kg (k6-:2a,)Q)=ōM1Qmfe  kō1۟Qmfe C  kg3>ލsandZ1=UR1yieldpSositivrerecurrentk#;ko2URN;suchthat@sk Wwg x!8"(0;1)&andN k Wwg x!"1 :sAgainthe rstconrvergenceisclear,whilethesecondonefollorwsfrom@sPk#(W<URkg)-TPk#(Y1:::Yk62 <URkg)-TPk#(S UU1ik62(\fYi,=UR0g)+Pk(T UU1ik62(\fYi,=ōM1Qmfe  k ?Cg)#4=T1(1ō1۟Qmfe C  kg3T) k6-:2 +(1ō/1۟Qmfe  k ) k6-:2 !UR0:s9.Furtherergo`dictheoremssThe!meanergoSdictheorem(7.1)canbestrengthenedtowreakconvergenceinstheJnpSositivrerecurrentcase,jyholdinginanextendedsensealsoinbSoththeotherscases.-oThis<-isaconsequenceofmoregeneralresultsin[18],PbutforthestatesspaceR+ IadirectproSofistoosimpletorefertootherwrork.=\Inthepresent@8 o썠:-iӍ}-sframewrorkthefollowingholds,indepSendentlyoftheinitiallaw:s(9.1)Theorem.If35n:=URL(XnP)forn0,thens(a)@wkn ܽwg !R;swhenever35ispffositiverecurrentwithstationarydistribution,s(b)An ܽwg !R"1 ;i.e.3<nP([0;t])UR!0for35all+Yt<1;swhenever35isnullrffecurrent35ortrffansient.sPrffoof.(a)BybSoundedconrvergence(2.6)implies@snPffQ=URE(qīR(fG(Xn)fG(X xڍn:j))(dx))UR!0forall9ZfQ2K,`(R+x):.(b)Itfollorwsfromtherepresentation@sW=URZ1j+:::+Y1:::Yn1(ZnR+YnPZn+1/t+:::uH)for)n2Nsthat:theevrent:fW:=t1gisconrtainedin(thecompletionof8)thetail{ eldofs(YnP;Zn)n2HN,bSecauseP(Y0=e0)=0bry(8.3). ThereforeW6+=1a.s.Dbry(8.2)sandthisimpliesbrymonotonicity@snP([0;t])w:XP(X 0ڍn URt)w=:XP(WnURt)v:V!:XP(WURt)=0forall9Zt<1: 2.ClearlyV,(9.1)strengthens(7.6)inthepSositivrerecurrentcase.uMoreover,sthecondition\X nite{vXalued"in(1.4b)cannorwbSereplacedbytheconditionsP(XF=UR1)6=1,kbSecausetconrvergenceinprobabilityimpliesweakconvergence..ThepSoinrtwiseergoSdictheoremforthepositivrerecurrentcasecanbSederivedsfromz(7.4). TVoobtainitforasmanryfunctionsaspSossible,itis,horwever,spreferable*toestablish rstergoSdicitryunderstationarityV.Actuallyastrongersresultholds:s(9.2)Prop`osition.If35theprffocess35(XnP)n0isstationary,itismixing.sPrffoof.1.ExtendingB^(YnP;Zn);nUR2N;let(YnP;Zn);nUR2Z;bSeindependenrtwithsdistribution.8Thenbry(8.2)@sX 0ڍn:=URP㋟1:::$Yn<UR1a.s.sand,moreorver,UR=L(X20RAnP)isthestationarydistribution.8Since@sX 0ڍn=URX 0ڍn1YnR+ZnPfor.*n2N;sthe2!proScesses(XnP)n0and(X20RAn)n0harve2!thesamedistributionanditsucesstoprorvetheassertionfor(X20RAnP)n0.@9 }썠:-iӍ}-.2.Denote}bry andn920theshiftinQDn2HZ"amR22RA+ andinQDn0!R+x,respSectivelyV,sandconsiderthemapping@so:UR(ynP;zn)n2HZ}!(P 91:::$yn)n0sfromꨟQ\on2HZ"R22RA+  toꨟQ\on0!9R+x.8ItiscompatiblewithXandn9200itcanbSeshorwnthat(9.2)holdsinthestrongerssensezof(XnP)n0harvingatrivialtail{ eld. Thiscanfailextremelyinthesgeneralcase,asisseenbrythefollowingexample:@sY=UR1=2and<,P(Z1=0)=1=2=P(Z=1)sobrviouslyUrepresentsthesimplestnontrivialpSositiverecurrentcase. yxHeres(XnP)n0is7stationaryV,iftheinitiallarwistheuniformdistributionon[0,2].sSinceXn1otcanbSereconstructedfromXn via@sXn1=UR2(XnR1fXn7>1gr)a.s. ;sthetail{ eldof(XnP)n0vcoincidesinthiscasewiththefulln9{algebrageneratedsbry(XnP)n0ot(moSdulonullsets)..Norw(thepSointwiseergoSdictheoremisasimpleconsequence.صRelatedresultssin7theconrtextofLipschitzmapscanbSefoundin[3],[10],[11].LIn7thepresentsframewrorkthefollowingistrue,againindepSendentlyoftheinitiallaw:s(9.3)Theorem.Lffetyw@bepositiverecurrentwithstationarydistributionsand35f{4bffe{integrable.fiThentheconvergence/JōB81AQmfe  nKԟPXl 0m05theconditions(a)and(b)canbSeshorwnstoM<22k ʟimpliesXn+1<22k6+1,&theysatisfyT1 V0g:sSinceAk :isa nitesubsetof@sBkx:=UR[2 k#;2 k6+1[ݿB11 {썠:-iӍ}-sand@thestationarydistributionisnonatomicbry(6.4),acontinuousfunctionsfQ:URR+ q!R+  canbSeconstructedsucrhthat捍s(2)fG(x)URkgnk#for.,lx2Ak#;wݍs(3)qīRBi?k)+fGdUR2 k for4ko2N:sThenbry(2)thesequence܍@sVn:=ōL1Qmfe  n?PMx1mn;~fG(Xm)for)nUR2Nxssatis es@sP(limsup&bn!1>Vn=UR1)& }P(limsup&bk6!1>&fVTi?k _IURkgg)& }P(limsup&bk6!1>&fTkxURnk#g);swherethelastinequalitryfollowsfromi0@sVTi?k _Iō1Qmfe   TkKfG(XTi?k )URkgfor)oTkxnk#:AsInviewof(1)thisimplies@slimsupfvIn!1Vn=UR+1a.s. ;swhilebry(3)ontheotherhandfQUR1..Therrestofthissectiontreatsmeanpassagetimes,where rsttheregionsabSorveRandbelorwtheinitiallevelwillbSeconsidered.nWhetherTisrecurrentorstransienrt,theascendingladderindiceshavea niteexpSectation:s(9.4)Prop`osition.Lffetg}TB denotethe{possiblyin nite{hittingtimeofasset35BX2URB]m(R+x)by(XnP)n0.fiThen,forarbitrffaryx<&feR]ڍx ,@sE xH(T[x;1[K)UR<1:sPrffoof.DuetoxUR<&feR]ڍx LthereexistsnUR2Nsucrhthat@s#UR:=P(X 0ڍn x)>0:sSincebrymonotonicity@sfT[x;1[3>URkgngufXn<URx;:::ʜ;Xk6n 4URkgn)(P(X 0ڍn xfeR ,ls(a)@wkE xH(T[0;x]G)UR<1wRhenevn9eris35positiverSecurrent;s(b)AE xH(T[0;x]G)UR=1wRhenevn9eris35nulClvrSecurrent:sPrffoof.(a)1LetbSethestationarydistributionandL(X0)UR=.cThen1itfollorwssfromtherecurrencetheorembryKac[21]that@sE(T[0;x]jURX0Vx)=(([0;x])) 1 \|:sSinceEbry(5.2a)thehittingdistributionon[0;x]equalsthedistributionofX0sunder.theconditionX0 lZVx,bryinductionthisequationcanbSeextendedtosthekg{thhittingtimeTk :of[0;x],i.e.@sE(TkxjURX0Vx)=kg(([0;x])) 1 \|:sThisimpliesallthatisneededinthesequel,namelys(1)qīRfXq0*xg?8lTk#dPUR<1forall9Zko2N:sNorwxUR<&feR]ڍx cmaybSeassumed,)!becauseotherwiseE2xH(T[0;x]G)UR=1bry(1.2b).=Understhisassumptionthereexistsko2URNsucrhthats(2)P(X0VURx0:sBytheinequalitry@sT:=URinfHfnUR>ko:XnxgTk6+1;swithx@0;k8Jdenotingthedistributionof(X0;Xk#),"andbrymonotonicityitfollowssthat@sqīRIfXq0*xgjSTk6+1 dPF8qīRFfXq0*x0x&similarlyGF(=]x ";x+"[F(R+ maryx&bSeassumed.ZChooses rstG20#=]xQl"209;x+"209[witharbitrary"20#2Qj]0;"[.*PInviewof(G209)UR>0itfollorwssasintheproSofof(9.5a)thats(1)E x-:q% cmsy60(TG0 _)UR<1forsomeFy|x 0#2G 09:sFVor xed!Ë2UR considernorwn2NwithX2x-:0RAn1(!n9)2G209.8Thentheinequalitry@sjX xڍn:j(!n9)X x-:0ڍn1(!)jO-=jxx 09jQo1mn.uYm(!n9)6E" 0ō21jQmfe {  x0Z(x 07Q?1mn0Ym(!n9))#4O-<" 0ōV1jQmfeԟ  x"0'(x+" 09) sinviewofjX2x-:0RAn1(!n9)xjUR<"20impliesthatص@sjX xڍn:j(!n9)xjUR<" 07(ōLx+"2033Qmfeԟ  x"0"+1)=ō2"209xQmfeԟ  x"0#a:ݿB14썠:-iӍ}-sThereforebry(1)э@sE xH(TG)URE x-:0(TG0 _)<1forō.c2"209x*< Qmfeԟ  x"0Ljf";ԍswherethelastconditionissatis edforsucienrtlysmall"209..2.8TVoprorvetheconversesuppSoseonly@sE xH(T[0;t]*)UR<1forall9Zt>x:sAssumingnorwonottobSepositivrerecurrentimpliess(2)P(Ypx+Z1>URt)=0forall9Zt>x:sIndeed,with1V:=URL(X2xRA1:j)monotonicitryyields@sE xH(T[0;t]*)wJqīRs>tNE sn<(1+T[0;t]*)1(ds)wJP(Ypx+Z1>URt)E t(T[0;t]*);swhereE2t(T[0;t]*)UR=1bry(9.5b).8Lettingt#xin(2)leadsto@sP(Ypx+Z1URx)=1;shencebry(1.2b)to&feR]ڍx4xIx<1.9ThisimpliespSositiverecurrenceandthusasconrtradictiontotheassumption: 2.TVogether,{(9.1)%and(9.6)shorwthatthetwomaincharacterizationsofspSositivre/null_recurrencefromclassicalMarkrov_chaintheoryinessencecarrysorvertoanerecursions.*s10.ThecontractivecasesAsoutlinedinSection5itisingeneralimpSossibletodeterminetheinrvXariantsmeasure<ofarecurrenrtdistributionSexplicitlyV..Inspiteof(8.2)thisholdssas9wrellforthestationarydistributioninthepSositiverecurrentcase(foransexceptionZsee(10.3)).}ThisiscompSensatedtosomeextenrtbythefactthat,sduedtothealgebraicformoftheunderlyingrecursion,themomenrtsofcansbSemeasilycomputedfromthoseof,asfarastheyexist.,wThefollorwingsimplescriterionfortheirexistenceextendsaresultbryVVervXaat[39]:s(10.1)Prop`osition.Lffet35(XnP)n0bestationaryand0UR< h<1.fiThen@sE(X ڍn)UR<1sif35andonlyif@sE(Yp )UR<1andu>0sif35andonlyif@sYUR1andu>0:sPrffoof.1.The.conditionsonYandZ 5arenecessaryV,?bSecauseE(X2kRAn)<1.forsallko2URNimpliesbry(10.1)@skYpk1 UZ=URlim k6!1-ʬkYkkxUR1;ݿB16m썠:-iӍ}-swhileE(e2uZ )UR<1isimmediatefromZ1VX1..2.5TVo3prorvesuciencyV,consider rstthespSecialcaseYUR#<1.5WithWsasde nedin(8.2)thisyieldstheestimateg֍@sE(e uW)pqE(exp*(uP7n2HN## n1ZnP)) Bp=qPhk60ōu2kQmfe Ӎ  ;kg!kP 9n2HN!# n1ZnPk kڍk%BpqPhk60ōu2kQmfe Ӎ  ;kg!(P 9n2HN!# n1kZnPkk#) k#4p=qPhk60ōw1Qmfe   kg!1(ō u33Qmfewn  1#) k#E(Zܞ k0)/p=qE(e uZ=(1#)& )for)uUR0;si.e.8ifuissuitedforZܞ,then(1#)uissuitedforW..3. TVo8reducethegeneralcasetothissituationapartitioninrtorandomsbloScrkswillbeused,Pwhicrhisdualtothatoneusedintheproofof(3.2).sApplyingP(Y+=;1)<1crhoSose#<1satisfyingP(Y+>#)<1anddenotesbry 0eA=T0 %E#g"#e uZm>^1fYn7#g \e uZnL)po=4P:mn2HN0( 1(u)) n1 2(u);_Ǎswhere@s 1(u)UR:=E(1fY>#g{e uZ )!P(Y>#)for)u!0;@s 2(u)UR:=E(1fY#g{e uZ )UPE(e uZ)forall9Zu0:ݿB17썠:-iӍ}-sFVoru0V>UR0with 1(u0)<1and 2(u0)<1thisyieldss(3)E(e uq0*Z-:0)UR 2(u0)=(1 1(u0))UR<1:sBy(1){(3)thereductiontothespSecialcaseissettled: 2.Theisimplestexampleforthesituationof(10.2)isprorvidedbythecases&feR]ڍx&<1.2IfinadditionthesuppSortNGis nite,5themeanrwhileclassical eldofsself{similaritry]isentered.Becauseoftheextensiveliteratureonthissubject,sinH"particularintheconrtextoffractals,onlyaquestionconcerninguniformsdistributionswillbSeconsideredhere:s(10.3)Prop`osition.Lffet35bepositiverecurrentwithxfeR5<UR&feR]ڍxand@sN6=URf(yid;zi):0ikggsbffe35suchthat0UR1,mconsideredindetailbyGarsia[15].qEvensfor@sN6=URf(0;1);(yn9;1)gwithō3 13 Qmfe  2=tsonlypartialresultsareknorwn,goingbackessentiallytoErd os[13]..TheGrestofthissectionisdevrotedtosituationsnotexcludingweakcon-stractions,s_i.e.toXthecaseY}1.The rstresultisofinrterestmainlyinthesconrtextof(10.6),thoughoSccursaslimitdistributionalsoinlearningtheorys(see8e.g. e[32])orinaproblemonrandomwralkstreatedbyMasimov[29]. eThesmethoSdfrom[13]yieldsthefollorwing:s(10.4)Prop`osition.With351UR6=l2Nand01_P(ZnP=log-GnUR>1)=P㋟n>1"hUP(e Zn >n)=1;swhicrhbyBorel{CantelliandXnURZn provesonehalfoftheequation..3.8TVoprorvetheotherhalf,choSoseanarbitrarytUR>1.8Then@sP(XnP=log-GnUR>t)o%=P(Zn>URtlog-Gn(  n1+Z1j+:::+  1g Zn1))V=E(exp(  n1+Z1j+:::+  1g Zn1/ttlog-Gn))sfornqQ>1,8bSecauseP(ZM>z)e2zbAholdsforz40aswrellandFVubiniappliessduetoindepSendence.8By(1)therefore@sP(XnP=log-GnUR>t)V=n t .@Q1m1;swherethein niteproSductisstrictlypositivrebyPxm2HN' 2m 2<UR1.8Thus@sPM n>1_P(XnP=log-GnUR>t)<1forall9Zt>1;si.e.81isalsoanuppSerbound: 2.In wconclusiontheorderinwhicrhanuppSerlimit&feR]ڍxK[=1isapproachedwillsbSestudiedforthewreaklycontractivemultiplicativemoSdel,beingofparticularsinrterest[inapplications.5WhiletheresultprovidesjustanuppSerboundinthescasePP(Y=UR1)=0,.itisexactotherwiseand,somewhatsurprisinglyV,depSendssonlyonthisprobabilitry:s(10.6)Theorem.If35YUR1andZ1=1,then@slimsupfvIn!1XnP=log-GnUR=1=logP(Y=UR1)a.s.sPrffoof.1. FClearlyV,X0 =1tucanbSesupposedinthesequel. FInprorvingthesrighrt{hand3sidetobSealowerbSoundfortheupperlimit,hmoreorver,YmaysbSedecreasedto1fY=1gAYp. ThereforeinthispartY$willbeassumedtobes0,W1{vXalued,where@spUR:=P(Y=0)>0:sThenHtherandomtimes0@=T0 iD gforanry H<1satisfyingP(Y=[ )>0,-bSecauseassubsequenrt#qlimitingproScedure ]!1yieldsthedesiredupperbound.:There-sforeBrestrictY tothevXalues and1and,with0replacedbry ,inrtroSducestheArandomvXariablesTk emandUkasinthe rstpartoftheproSof. >yNorwforsTk61U`URn1otexhausttheinrtervXal@sIF:=UR[0;1=log-GP(Y=1)]:sIndeed,-givrenlK2jNand"h>0,bryxfeR<1thereexistsarandomtimesTURl5_e21="׀sucrhthatXT=log-GT<UR",henceinviewof@sXn+1=log(n+1)UR(XnR+1)=log-GnXnP=logn+"for)nTsthevXaluesXnP=log-Gn;nURlC;are"{denseinI.ݿB22(ޠ썠:-iӍ}-sReferences"'K`y 3 cmr101.0sAlpuim,:M.:dAnextremalMark!oviansequence.xJ.Appl.Prob.,"V 3 cmbx1026,:219{232 0s(1989)"3.0sBarnsleye,>M.,Elton,J.,Hardin,D.:6Recurren!titeratedfunctionsystems.Con-0sstr.Appro!x.5,f3{31(1989)"4.0sBoro!vkov,WA.:tOntheergoMdicit!yandstabilityofthesequence( b> 3 cmmi10wzn+1s= f-(wznP;1zn).0sTheoryfProb.Appl.33,595{611(1989)"5.0sBrandt,A.:8ThestoMc!hasticequationYzn+1=)AznPYzn#+{`Bzn acwithstationarycoe-0scien!ts.Adv.Appl.Prob.18,f211{220(1986)"6.0sBrandt,kA.,Ferank!en,P.,Lisek,B.:=StationaryVkstoMc!hasticmodels.Chic!hester:0sWileyf1990"7.0sChama!you,,jJ.,Letac,G.:JExplicitstationarydistributionsforcompMositionsof0srandom$functionsandproMductsofrandommatrices.XcJ.Theor.Prob.4,D3{360s(1991) 10.0sElton,_1J.:An:IndianaUniv.Math.J.30,9v713{0s747f(1981) 21.0sKac,"M.:>OnthenotionofrecurrenceindiscretestoMc!hasticprocesses.n)Bull.0sAm.Math.SoMc.53,f1002{1010(1947) 23.0sKesten,H.:inThe.limitpMoin!tsofanormalizedrandomwalk.6Ann.Math.Stat.0s41,f1173{1205(1970) 24.0sKesten, H.:ӲRandomPdi erenceequationsandrenew!altheoryforproMductsof0srandomfmatrices.ActaMath.131,207{248(1973) 27.0sLev,w9G.:ŗSemi{Mark!ovCproMcessesofm!ultiplicationwithdrift. 9tTheoryProb.0sAppl.17,f159{164(1972) 29.0sMasimo!v,4V.:gAUgeneralizedyBernoullischemeanditslimitdistribution.9Theory0sProb.Appl.18,f521{530(1973) 30.0sMukherjea,eA.:vRecurren!t2randomwalkinnonnegativematrices:vattractorsof0scertainEiteratedfunctionsystems.zProb.TheoryERelatedFields91,|297{3060s(1992)ݿB234ؠ썠:-iӍ}- 31.0sMukherjea,BA.,TserpMes,N.:Measures#\ontopologicalsemigroups:con!volution 0sproMductsandrandomw!alks.2 Lect.NotesMath.547,4%Berlin{Heidelberg{New0sYeork:Springerf1976 32.0sNorman,M.:WLimitingdistributionsforsomerandomw!alksarisinginlearning0smoMdels.Ann.Math.Stat.37,f393{405(1966) 34.0sRac!hev,tS.,SamoroMdnitskye,G.:iLimitg}lawsforastoMchasticproMcessandrandom0srecursionfarisinginprobabilit!ymoMdelling.Preprint(1992) 39.0sVeervdDaat,W.:gOnaastoMc!hasticdi erenceequationandarepresentationofnon{0snegativ!e!jin nitelydivisiblerandomvdDariables.NAdv.Appl.Prob.11,@+750{7830s(1979) 40.0sWeolfe,S.: ˭OnNacon!tinuousNanalogueofthestoMc!hasticdi erenceequation0sXzn= %Xzn1+nBznP.StoMc!hasticfProcessesAppl.12,301{312(1982)ݿB24C;ʔ H2@cmbx8 3 cmmi10'K`y 3 cmr10#fcmti8K cmsy82cmmi8|{Ycmr8q% cmsy6;cmmi6Aacmr6u cmex10H2