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SIZE-DEPENDENT BRANCHING CHAINS IN RANDOM ENVIRONMENTS

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SIZE-DEPENDENT BRANCHING CHAINS IN RANDOM ENVIRONMENTS

    SIZE-DEPENDENT BRANCHING CHAINS IN RANDOM ENVIRONMENTS Availableonlineatwww.sciencedirect.com

    

    ScienceDirect

    ActaMathematicaScientia2010,30B(4):10651072

    数学物理

    http://actams.wipm.ac.ca

    EXTINCTIoN0FP0PULAT10N.SIZEDEPENDENT

    BRANCHINGCHAINSINRAND0M

    ENVIR0NMENTS

    WangWeigang(王伟刚)

    SchoolofStatisticsandMathematics,ZhejiangGongshangUniversity,Hangzhou310018,C

    hina

    Email:wwgys2000@163.corn

    Li}ran(李燕)

    DepartmentofMathematicsandComputerScience,ChinaUniversityofPetroleumJDongyi

    ng257061,China

    HuDihe(胡迪鹤)

    SchoolofMathematicsandStatistics,WuhanUniversity,Wuhan430072,China AbstractWeconsiderapopulation-size.dependentbranchingchaininageneralrandom environment.Wegivesufficidentconditionsforcertainextinctionandfornoncertainex

    tinction.Thechainexhibitsdifferentasymptoticaccordingtosupk. 0mk,

    0<1,mk,

    0'_1

    as七一_+?,n__..,infk,

0mk,

    0>1.

    KeywordsStochasticpopulationmodels;branchingchainsinrandomenvironments; extinctionprobability

    2000MRSubjectClassification60F08

    0Introduction

    TheGalton-Watsonbranchingchainisabasicstochasticmode1forapopulationofparti

    clesdevelopingintime.Thismodelwasgeneralizedtovariousbranchingmodelssuchasthe population-size

    dependentbranchingchain,wherethelawofoffspringdistributiondependson thepopulationsize,whichwasconsideredin14].Hu[5

    studiedtheexistenceofthecanonical

    branchingchaininrandomenvironment.WangandFang6-8studiedthepopulation-size

    

    dependentbranchingchainsinrandomenvironmentswhentheenvironmentsareindependent,

    stationary,orMarkovchain.Inthisarticle,westudyapopulation-size

    dependentbranching

    chaininageneralrandomenvironment.

    ReceivedDecember21,2006;revisedOctober9,2008.ResearchsupportedbytheNationalNatural

    ScienceFoundationofChina(10771185,10926036)andZhejiangProvinicialNaturalScienceFoundationof

    China(Y6090172).

    1066?TAMHEMATICASCIENTIAVoI.30Ser.B

    1DescriptionoftheModel

    Let(Q,,P)beagivenprobabilityspace.Let(O,E)beameasurablespace?Weintroduce thenotations={.,kin)and=foranyrandomsequenceXi,i=0,1,2,.

    Let{(pk,(?))0,0?

    O)beafamilyofsequencesofprobabilitydistributiononthenon-negative

integers,satislyingtheconditions

    +oo

    ?ipk()<..,0<Pk,(0)+Pk,oO)<1,V?,0?0i=0

    Let={()}obeasequenceofrandommappingsfrom(Q,,P)into(0,?).Forany

    0?0andanynonnegativeintegerassociatetheprobabilitygeneratingfunction

    ,

    (s)=?Pk,()s,0sl

    i=0

    Let={)0withZo=1beanonnegativeintegervaluedrandomsequencedefinedon (Q,,P).If:{)0satisfiestherelation

    E(sziz,(s)]zn,0s1,=0,1,2,,(1)

    then.wesaythatisapopulationsize

    dependentbranchingprocessinarandomenvironment. Then,iscalledtheenvironmentprocess.Conditionedon=wealsousethefollowing

    representationf0r

    Z

    +=

    ?

    J=1

    (2)

    whereforeach他?Nconditiononn=,therandomvariables{;}J?Narei.i.d.indepen- dentofwithP.g.f.,0(s).

    Inthisarticle,withoutlossofgenerality,wealwaysassumethatZ0=1.

    Letmk,0=(1)and%2,

    =

    .(1)+m,.一仇2bethemeanandthevarianceofthe

    offspringdistribution,respectively,whenpopulationsizeisequaltokandtheenvironmentis

    0.Let

    Q={;n()=0forsome.n),q=P(Q),andq(?)=P(QIO.

    WerefertoQasthesetofextinctions,whileqandq(?)are,respectively,theextinction-+ probabilityandtheextinctionprobabilityconditionedon?.Itisobviousthatq=Eq(()).

    2ExtinctionProbabilities

    Lemma1Supposeliminfpk,?(1)<1,V?N.Then,ifthelimitofZnconditionedOil.n?)

    ?existsalmostsurelyasrt__?..,then,

    )=1.a.Sn}.. P(1imZn=0oro.l?

    ProofWeonlyhavetoprovethat.foranyk?N+

    Pflimf0a.s

    No.4Wangetal:EXTINCTIONOFPOPULATIONSIZEDEPENDENTBRANCHINGCHAINS1067 Bylin1infpk,?(1)<l,wecangetHI<n2<<TLis.t.Pk,<co<1,Vi1.Becausen)

    Zisanonnegativeintegervaluedrandomsequence.wehave

    P(1imZn

    +..

    P(n{

    From(2),wehave

    .lf)

    )l)=P(

    P(+1=l=,f)

    S0,P(+1=lZn,?)

    thatthispolynomialisf(xo,Xl, +..+?

    P(Un{)l?)

    l=,?)

    ?Pk(1)pk(2)Pk()

    i10,i2?O,,ikO,

    il+i2++ik=

    apolynomial

,xk),thatis,

    P(z+1=lz=,?)

    ofPk,

    ?(0),Pk,?(1),???,Pk,?().We

    f(pk,?(O),Pk,?(1),'?',Pk,?())

    Byf(xo)Xl,?--,Xk)isapolynomial,wecangeta0,al,?,ak,s.t f(ao,al,-,ak)supf(Xo,1,?,z)=Po

    xi0,Vi?0;x1<cO,

    XO+X1+???+Xk=1

    aSSUme

    Itisobviousthata0+al+-?-+ak=1,0<P0<1,thatisbecause,ifletX,m,m

    l,2,,arerandomvariablesand{):1bei.i.d.s.t.X

    a0,P(X11)=alco,,P(X1k)ale,then,P(=k)=P0

    P(+1=l.=,?)po

    So,from(3)and(5),wehave P(1imZ

    Thelemmaisproved.

    Remark1NotetheconditionofLemma1

    ?X,P(X1=0)

    m=1

    hence,

    t}lOIIPo=0

    }?)=0

    lirai

    ..

    nfPk,?(1)<1,Vk?Nlim

    i

    ..

    nfPk,?(1)?1,Vk--

    +

?N??

    n..

    (5)

    InthemodelofGalton

    .Watsonbranchingprocessorpopulation--sizedependentbranching

    process,wealwaysassumethat0<p(0)+p(1)<1or0<pk(0)+pk(1)<1,Vk?N.So, theconditionhereisnotstrange. Inthefollowingofthisarticle,wealwaysassumethatlimsupPk,0(1)<1.

    n—?..

    Theorem1Iflirasupsupmz,

    0<

    n—?1?N+

    P(1imZnol{

    +

    Z

    P

    n

    1_

    ,.L

    n

    P

    <

    ,?,

    _l

    ,?Ln

    P

    1068ACTAMATHEMATICASCIENTIAVl01.3OSer.B ProofByhypothesis,thereisanN>0,s.t.Vn>N,supm1.

    o<a<1.So,wemight

    IcN+

    aLswellassumethatsupml, oa<1.

    ZEN+,hEN

    Because(s)isanondecreasingfunction,wehave

    Hence,

    S0.

    0;,(s);,(1)<1,Vl?N+,n?N,s?f0,1]

    l,0ns)f,0n(0):c,0(8)pf,0(0)=,()sas,77?(0,s), ,0(s)?a8+pf,0(0)as+1

    E(()f,:=()(.+)=(),

    whichshowsthat{\11n,"~ZnJln+:ooo thereisafinitevariableVs.t. conditionedon?=0isanon-negativesupermartingale.Hence,

    ByLemma1and(6),weobtain /1,J

    P(1ira=0=0)=1n_?OO

    (6)

    Ihetheoremisproved.

    Remark2ByPl,0(1)ml,0,Vl,礼?N,wecanalsoverifythattheconditionoflemma

    1iStruehere.

    +oo

    Theorem2Letn=supp2,?.If?.n<o.,then,g()<1.a.s..

    ProofBecause

    So,wehave

    +o.

    P(=0J=P(U{=o}16=1O0n—一

    =1

    Hence,foranyk>0,

    P(1im

?0)J),

    P({Zn+1?0}I{?0)),

    P({+1?o}Iz~,西=1p(0) P({Zn+1?0)1[?0},10

    =

    0I)1P({Zk?o}l,qa) +o.+o.

    Bythehypothesis?a<..,wecangeta>0,suchthat(1a)>0.So,

    n=ln=k+l

    g()=P(1im=0i)<1n+..

    n

    0

    *

    Z

    r?L

    P

    ll

    ?

    r

    0

    *

    ,?L

    n

    P

    No4Wangeta1:EXTINCTIONOFPOPULATIONSIZE

    DEPENDENTBRANCHINGCHAINS1069

    Thetheoremisproved. Lemma2[9]ForanonnegativerandomvariableX,wehaveE(1)

    where,(s)Esx,0s1. \f(s)ds

Theorem3Supposethat(s)ds1,Vk?+,n?N.Then,q(<1.spe

    cially,ifl,(s)ds1,V?N+,a.s.0?O,wehaveq<1. ProofFrom(1)andLemma(2),wehave E(z,?厂1,,/,8(s)ds?0

    whichshowsthat{)c.nditi.ned.n:6isanonnegatiVesripermartinga1e,andSO

    convergesalmostsurlytoafinitevariable.From(7),weobtain

    E(=E(:/

    N.w,byLenlma1anddominatedc.1wergencetheorem,E(j==q(,q(<1

    S.,ifl,(s)ds1,V?N+,a.s_0?O,wehaveq(0<1,Paandhence, qEq()<l

    'hetheorem1Sproved.

    Inthefollowingtheorems,weshallbeconcernedonlywiththecase{m,)isanon increasingsequenceVn?,m,

    m=1,andtheconvergenceisuniformwithn?N

    ;,<?.DenoteE,0=m,.l,?N+,n?N.

    Theorem4If{m,)-t-=c~1isanonincreasingsequence,m,1,弧竹>0,Vn?N then,

    P(?,n=0

    ProofWehavefrom(2)

    Therefore

    ..?国]

    E(Zn+1l,==Znmz=Zn(1+cz0)

    ,,,

    l

    ,?=l

    }1(1+C

    Z,0k)

    k=0

    whichshowsthatf_L}conditionedon=isanonnegativemartingale,hence,L(1+

?z,0)

    k=0

    n1

    coI1vergesalmostsurelytoafinitevariable.As(1EZk,)isnon

    decreasing,SOthelimit

    of(1+EZ,0k)a.S.exists. k=0

    n1

    (1+E,0)__?..a,S. Wewillprovethat

    1070ACTAMATHEMATICASCIENTIAV01.30Set.B

    礼)1

    Supposen(1

    k=0

    Ek)_n,0<.<.o.WemnstVerifyth,exists.Hence,by

    Lemma1,limz=OOonQ.Thisiscontradictno.

    wi{Z~+l,

    tconvergingtoafinite

    v{iab1e.Hence,thesupp.seisfalse.(1+EZk,Ok)..a.s..n.Hence, k=0

    ..=?Q)=1

    Thetheoremisproved.

    Corollary1Ifli(1)<l?N,<o.then, n

    q+?

    ——

    ?.o"=f1

    Weshallassumethefollowingconditions:

    (A).1imk~k,0=c,0SC<O(3

    o.

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