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The Moments of First Entrance Time Distributions of Markov Chains in Random Environments

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The Moments of First Entrance Time Distributions of Markov Chains in Random Environments

    The Moments of First Entrance Time Distributions of Markov Chains in Random

    Environments

    Chin.Quart.

    2011,26(1):

    3.ofMath

    51——55

    TheMomentsofFirstEntranceT

    DistributionsofMarkovChains

    RandomEnvironments

    LIShouwei,ZHUDong-jin2

    lme

    ln

    (1.SchoolofEconomicsandManagement,SoutheastUn!versity,Nanjing211189JChina;2.Department

    o/Mathematics,AnhuiNormalUniversity,Wuhu241003,China)

    Abstract:Afterdefininggeneratingfunctions,thispaperdiscussestheirproperties,and thenprovidesasufficientandnecessaryconditionforafinitepropertyofthemomentsof firstentrancetimedistributionsofMarkovchainsinrandomenvironmentsbygenerating functions.Finally,thepaperobtainsrelevantconclusionsofthemomentsoffirstentrance timedistributiOns.

    Keywords:randomenvironments;tabooset;generatingfunctions

    2000MRSubjectClassification:60J10.60F05

    CLCnumber:O211.62Documentcode:A

    ArticleID:1002?0462(2011)01005105

    ?1.Introduction

    MarkovchainsinrandomenvironmentsiSanaturalgeneralizationofclassicalMarkovchains.

    Before1980'Smanyimportantachievementsonthissubjecthadbeenmadeandthesetheories mainlystudiedsomeparticularmodels,suchasbranchingprocesses,randomwalkinrandom environments,etc.Cogburnlestablishedageneraltheoryforthistopic,andsuccessfully studiedthepropertiesofMarkovchainsinrandomenvironmentsintheframeworkofHopf Markovchains.Hehasmadedmanyimportantfindingsinthisfield,suchastheclassifica- tionofstates,invariancemeasures,ergodiclimittheorems,centrallimittheorems,andSOon. Orey[dealtwiththeergodictheoriesoftheseprocessesandproposedaseriesofopenques. tions.Lit,71provedtheexistence,introducedtheconceptsof7r

    irreducibility,recurrenceand

    Receiveddate:2007.11-08

    Foundationitem:SupportedbyNSFofAnhuiProvince(KJ2007A102)

    Biography:LIShou-wei(1984?),male,nativeofBengbu,Anhui,Ph.D.,engagesinmarkovchainsinrandom

    environments.

    52CHINESEQUARTERLYJOURNALOFMATHEMATICSVb1.26

    transienceforMarkovchainsinbi

    infiniterandomenvironmentsandpartiallyansweredOrey's

    openquestions.Theirpaperscontainnumerousreferencestoearlierrelatedwork.Inthispaper,

    wemainlydiscussthefinitepropertyofthemomentsoffirstentrancetimedistributionsbythe generatingfunctions.

    ?2.BasicNotations

    LetNdenotethesetofnonnegativeintegers,?+denotethesetofpositiveintegers,andZ

    denotethesetofintegers.Let(,,P)beaprobabilityspace,beadenumerablespaceand AthediscretefieldsofX.Let(e,)beanarbitrarymeasurablespace.Let={,n?z)

    and={,佗??)berandomsequenceson(Q,,P)respectivelytakingvaluesine

    andX.Let{P():0?e)beafamilyoftransitionfunctionsdefinedin(X,A).Assume

    thatP(?;?,A)isB×AmeasurableforarbitraryA?A..Weset={n,kr,一? r?)foranarbitrarysequence'=.[n..Let={n,n?z>=(,1,00,1,), whichisacOordinateprocessdeflnedinezLet

    J

    ~rdenotethedistributionofprocess definedinBzandTbethecoordinateshift,i.e.Tk=.[,n),=On+%,k=,-1,0,1. LetP(Oj)denotetheproductoftransitionmatricesP(Oj)P(Ok).Letdenotea

    countingmeasuredefinedonA,WesetE=X×ez,?:AxBz,and:×7r.Definea tr(,?):,;{)×)=,)B().??,(F)={,ans

    :

    it

    (

    io

    ,

    nfu

    )

    n

    ?

    ct

    F

    ion

    ),

    i

    [

    n

    F

    E

    ={(

    P

    ,

(x

    ):

    Y

    ?(F

    B

    )).

    p(0o;ForF

    P

    w

    "

    e

    (

    se

    ,

    t

    Thenthegeneralformulais;F): ?p(Oo…一l;z,)F)(T").

    yEx

    Definition1IfVA?A,n?N:

    (1)P(Xo?AI-6

    (2)P(X+?A

    environment?.

    :P(Xo?Al0-..);

    ,):P(;Xn,A),theniscalledaMarkovchaininrandom

    LetHcXbeanarbitrarysubsetofX.Weset=(n,T),n0,p(n(;z,)= f(?{)xOz,Hzoz,1k<n)whereHCXiscalledatabooset,

    ,(n(;,)=P(,)(?{)xOz,讥隹{f)xOz,1<),,(n)(;,)=(,)(? <)xOz,叼隹(HU{y})xOz,1k).IfH={),w.ewriteHp()(;,Y)=p((;,).

    IfH={)UH1,wewritep(n(;,)=,1p(n)(;,暑『).Especially,,(n)(;,)=

    p(n)(;z,),H,(n)(;,!,)=,Hp(n)(;z,!,).Le1p(.)(;,!,)={z'y.),,zx?CHl,9(n)(;,)=?H,((;,),q(n(;,f)=?日9(%(;,).LetHmr(;z,):

    ?扎Hf(")(;,y)wherer?^r+,Hm(;,Y)iscalledthemomentoffirstentrancetime

    distribution.LetH,(;,Y)=?Hf()(;,).

    For8?(0,1),wedefinesomegeneratingfunctions. No.1LIShouweietal:TheMomentsofFirstEntranceTime

    Definition2

    F(,,;s):

    G(7,,;s):

    ?

    ?s"n-----1

    ?

    ?

    n----0

    ,(n(;,3,),

    8n9(n'(;,),

    c_.

    HQ(7,,;s)=?sn(n'(;,3).

    n----0

    ?3.PropertiesofGeneratingFunctions Property1IfHm(;,Y)<o.,thenallthefunctionsF(,,;s),HG(,z,,;s), Q(,,s;8)areinfinitelydifferentiablein(0,1). ProofWeonlyhavetoprovethedifferentiabilityofQ(,z,;8)in(0,1),sincethe proofforthdifferentiabilityofotherfun. ctionsissimilar.For.fixed80?(0,),let=

    

    ,then80<<1.Notethat(n)(;z,,)??9()(;z,s,)=m(;,暑『),and k>

    0

?nHm(;,Y)<oo.Since

    (sn日口(n'(;z,)),(s0):ns-.g(n'(;,暑『)?扎_.m(;z,), thenbydominatedconvergencetheorem d

    dsQ(,,;s)(8.)=?ns孑一1Hq(n'(;).

    n>1

    ThusQ(7,,;s)isinfinite1ydifferentiablebyinduction.

    Therelationshipsamongthesegeneratingfunctionsaregivenasthefollowing,whichare

    easilyobtained.

    Property2(1)If,?(;,,):1,then(1s)G(,z,;s):1一日F(,,3,;s). (2)(1s)HQ(7,z,;s):m(;,)一日G(7,,3f;s).

    ?4.TheMomentsofFirstEntranceTimeDistributions

    The0rem1If/-/m(7:z,3,;s)existsandisfinite.,)<?,thenmr+.(;,) ProofFirstweprovethenecessity. ?n>r7')18n--rHg(n'(;,3,)

    <c~ifandonlyif

    .

    l

    _+

    im

    l

    dr

    H

    Q(7,

    54CHINESEQUARTERLYJOURNALOFMATHEMATICSVb1.26

    ?

    n>r

    (佗一r)!(n(;,)

    m>k

    r+2,(m(;,):mr+.(;z,). WeobtaLinQ(-8,,;s)existsandisnitebymr+.(;,)<?.

    ?

    n>r

    Q(,,;s)

    !

    

    r)!(n(;,)

    c

    ?日9((;x,y)kr+1 k>_

    rT1

    =c

    ?日,(m(;,)(?r+1一?kr+1) >C

    m>r+2

    ?1?<

    ,(m(;,s,mr+2c

    mr+2

    :cmr+(;,)c?

    1<k<r+1 ?1<k<r,(m(;,) m?r+2

    kr+2(;,)c?H,(m(;,). Sincem(;,)<?,thenc?r+ 1r+l

    Thereforethesufficiencyisright.

    m>r+2

    ,()(;,)+c?日,(m)(;z,)isfinite.

    mr+2

    Wewrite-?Yincase7r{:,(;z,Y)>0)=1. The0rem2If7risstationary,-?Y,,(;z,z)=17ra.s.,mr(;,)<?不一a.s.,——.——————?

    then

    m(;,Y)<?7ra.s..

    ProofFirst,wecaneasilyobtain mr(;,z)mr(;,)+o.

    "=1

    (;,)

    Since7risstationary,bytheProposition2.11[wehave Consequentlywehave

    Therefore,

    ,(;,):17ra_s_.

    ?

    ?8=1

    Vn0,,(T;Y,)=17r?a.s.._'_

    mr(;,)mr(;,)+

    (;,)

    mr(;,).

    l1?HU

    NO.1LIShou.weietal:TheMomentsofFirstEntranceTime55

    Hencethetheoremisobtainedbym(;z,)<?7r-a.s..-?一?

    Ifweenumeratethestatesinanywaywithpositiveintegers.thenwesay_?OOifthe integercorrespondingtothestatetendto(x3.

    The.rem3Ifforsomer?1,mr(;,)<?,then

    .

    1im,.

    .m(;X,Y)=m(;X,Y)--

    +?

ProofReplacingHby{)in(1)oftheTheorem1[9jwehave

    ,(n'(;z,)::,(n(;,)+n?--I,((;,),(n"'(;,).

    Thenforarbitraryn?1,0,(n(;,),(n(;,)n?--1,()(;

    ,).Since

    -

    ~-oo

    ,(")(;z,z)=1,thenf0rarbitraryv,wehave,((;,)=.. Consequently,wehavelim,(n(;,)=,(n(;,).SinceZ_(9O

    , ?nr,(n'(;,)mr(-Lz,)<?

    n=1

    bydominatedconvergencetheorem

    m(;,)=-

    i

    -

    +?n.,('(;,)=Z+oo:o.'n:=1

    [References]

    o.

    ?

    n=1

    n,(n(;z,):mr(;,)

    [1COGBURNR.Markovchainsinrandomenvironments:thecaseofMarkovianenvironment

    [J].AnnProb,

    1980,8(3):908916.

    [2COGBURNR.TheergodictheoryofMarkovchainsinrandomenvironments[J].ZW,1984,6

    6(2):109128.

    [3]COGBURNR.OndirectconvergenceandperiodicityfortransitionprobabilitiesofMark

    ovchainsinrandom

environments[J].AnnProb,1990,18(2):642654.

    [4]COGBURNR.OnthecentrallimittheoremforMarkovchainsinrandomenvironments[J].AnnProb,1980,

    19:587.604.

    [5

    OREYS.Markovchainswithstochasticallystationarytransitionprobabilities[J].AnnProb,

    1991,19(3)

    907.928.

    [6LIYing-qiu.RecurrenceandinvariantmeasureofMarkovchainsinbi

    infiniterandomenvironment[J]

    ScienceinChina,2001,44(10):12941299.

    [7LIYing-qiu.TransienceandinvariantfunctionsforMarkovchainsinbi

    infiniterandomenvironment[J

    ChineseAnnalsofMath,2003,24(2):515-520.

    [8]8HUDi-he.FromP

    mchaintoMarkovchaininrandomenvironment[J].ChinAnnMath,2004,25A(1): 65-78.

    9]9LIYong?kui,HUDi-he.ThetabooprobabilityofdiscreteMarkovchaininrandomenvironments[J].Wuhan

    Univ(NatSciEd),2006,52(3):273?276.

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