季金莉,　石志巖,　楊衛(wèi)國(guó)

(江蘇大學(xué)理學(xué)院, 江蘇鎮(zhèn)江212013)

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關(guān)于樹(shù)指標(biāo)二階齊次馬氏鏈的等價(jià)定義

季金莉,石志巖,楊衛(wèi)國(guó)

(江蘇大學(xué)理學(xué)院, 江蘇鎮(zhèn)江212013)

[摘要]樹(shù)指標(biāo)隨機(jī)過(guò)程是近年來(lái)概率論的研究方向之一，已引起了概率論、物理學(xué)、計(jì)算機(jī)等學(xué)科的廣泛關(guān)注，國(guó)內(nèi)外關(guān)于樹(shù)指標(biāo)隨機(jī)過(guò)程的研究已經(jīng)取得了一定的成果.樹(shù)指標(biāo)隨機(jī)過(guò)程中的一類重要的模型就是樹(shù)指標(biāo)馬氏鏈.Benjiamini和Peres首先給出了樹(shù)指標(biāo)馬氏鏈的定義.楊衛(wèi)國(guó)、陳曉雪和王豹給出了樹(shù)指標(biāo)一階馬氏鏈的等價(jià)定義.楊衛(wèi)國(guó)等又研究了樹(shù)指標(biāo)馬氏鏈強(qiáng)極限定理.為了更有效的研究樹(shù)指標(biāo)隨機(jī)過(guò)程，本文給出樹(shù)指標(biāo)二階齊次馬氏鏈的等價(jià)定義，并證明其等價(jià)性.

[關(guān)鍵詞]等價(jià)定義; 樹(shù); 馬氏鏈; 樹(shù)指標(biāo)隨機(jī)過(guò)程

1引言

圖1　Gayley樹(shù)TC,2形成的雙根樹(shù)

樹(shù)指標(biāo)隨機(jī)過(guò)程是用來(lái)描述具有樹(shù)的結(jié)構(gòu)的數(shù)據(jù)模型，在生物學(xué)中有著很好的應(yīng)用.生物學(xué)家研究桿狀菌的分裂時(shí)，總結(jié)出桿狀菌分裂的規(guī)律，即一個(gè)桿狀菌在分裂時(shí)，從中間斷開(kāi)，這樣分裂成兩個(gè)新桿狀菌，這兩個(gè)新的桿狀菌為原來(lái)的桿狀菌的后代.如果我們把每一次分裂中的桿狀菌看成一個(gè)頂點(diǎn)，那么所有桿狀菌全體就是一個(gè)樹(shù)指標(biāo)隨機(jī)過(guò)程[1].因此研究樹(shù)指標(biāo)隨機(jī)過(guò)程的各種極限理論不僅有較高的理論意義，同時(shí)也具有較好的實(shí)際應(yīng)用價(jià)值.Benjiamini和Peres[2]首先給出了樹(shù)指標(biāo)馬氏鏈的定義并研究了其常返性質(zhì)和射線常返性質(zhì).Berger[3]等研究了齊次樹(shù)圖上平穩(wěn)隨機(jī)場(chǎng)熵率的存在性.楊衛(wèi)國(guó)等[4-6]研究了齊次樹(shù)圖上馬氏鏈場(chǎng)狀態(tài)發(fā)生頻率，以及二階有限非齊次馬氏鏈的強(qiáng)大數(shù)定律.楊衛(wèi)國(guó)等[7]研究了樹(shù)指標(biāo)馬氏鏈的等價(jià)定義.石志巖等[8]研究了樹(shù)上二階非齊次馬氏鏈隨機(jī)轉(zhuǎn)移概率調(diào)和平均的極限性質(zhì).本節(jié)我們主要給出樹(shù)指標(biāo)二階齊次馬氏鏈的定義，并研究樹(shù)指標(biāo)二階齊次馬氏鏈若干的等價(jià)定義.

則稱P為二階轉(zhuǎn)移矩陣.

p=(p(x,y)),x,y∈G,

(1)

(2)

分別為G2上的概率分布和G3上的二階轉(zhuǎn)移矩陣.如果對(duì)于任何頂點(diǎn)t,

(3)

且

P(X-1=x-1,X0=x0)=P(x-1,x0),?x-1,x0∈G,

(4)

2主要結(jié)果

給出二階樹(shù)指標(biāo)馬氏鏈的等價(jià)定義.

(5)

證(i)→(ii)：由(i)式及條件概率的乘法公式有

=P(X-1=x-1,X0=x0,XL1=xL1,XL2=xL2…XLn=xLn)

(6)

設(shè)i≥1,?t∈Li,有

=…

將上式代入(6)中

顯然有

P(X-1=x-1,X0=x0)=P(x-1,x0),x-1,x0∈G,

只需要證明，對(duì)于任何頂點(diǎn)t,(t≠-1,0)

(7)

又

P(X1t=x1t,X2t=x2t,Xt1=xt1,Xt2=xt2,…,Xtn=xtn)

(8)

于是得

即(i)式成立.

注2將(5)式作為二階齊次馬氏鏈的等價(jià)定義.如果t僅和父代1t有關(guān)，就是陳曉雪[7]等研究的樹(shù)指標(biāo)馬氏鏈等價(jià)定義.顯然本文t不僅和父代1t有關(guān)，而且和祖父代2t也有關(guān)系，所以本文將樹(shù)指標(biāo)馬氏鏈等價(jià)定義的推廣到樹(shù)指標(biāo)二階馬氏鏈.類似地，也可以給出樹(shù)指標(biāo)高階馬氏鏈的定義.

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The Equivalent Definition of Second-Order

T-indexed Markov Chains

JIJin-li，SHIZhi-yan，YANGWei-guo

(Faculty of Science, Jiangsu University, Zhenjiang 212013, China)

Abstract:T-indexed stochastic processes have been one of the research directions in probability theory,which have draw wide attention of probability theory,physics,computer science and so on in recent years.There have been some works on tree-indexed stochastic processes at home and abroad.Tree-indexed Markov chain is a kind of important model of tree-indexed stochastic processes.Benjiamini and Peres firstly give the definition of Markov chains.Yang,Chen and Wang give the equivalent definition of t-indexed Markov chains.Yang have studied the strong laws of large numbers and Asymptotic Equipartition Property (AEP) for Markov chains field on trees and studied the strong limit theorem of t-indexed Markov chains.In order to study series of related problems about t-indexed stochastic processes efficiently,this paper presents the equivalent definition of second-order t-indexed Markov chains and proves the equivalence of it.

Key words:equivalent definition; tree; Markov chains; T-indexed Markov Chains; tree-indexed stochastic processes

[基金項(xiàng)目]國(guó)家自然科學(xué)基金資助課題(11071104,11226210); 江蘇大學(xué)高級(jí)人才啟動(dòng)基金(11JDG116)；江蘇省教育廳統(tǒng)計(jì)應(yīng)用研究基地；2014年江蘇大學(xué)統(tǒng)計(jì)學(xué)校級(jí)重點(diǎn)學(xué)科

[收稿日期]2014-11-20

[中圖分類號(hào)]O211.6

[文獻(xiàn)標(biāo)識(shí)碼]A

[文章編號(hào)]1672-1454(2015)02-0010-04