隱非齊次馬爾可夫模型的強大數(shù)定律

吳小太,吳艷蕾

(安徽工程科技學(xué)院數(shù)學(xué)系,安徽蕪湖 241000)

隱非齊次馬爾可夫模型的強大數(shù)定律

吳小太,吳艷蕾

(安徽工程科技學(xué)院數(shù)學(xué)系,安徽蕪湖 241000)

在狀態(tài)集都有限的情況下,給出了隱馬爾可夫模型的一些性質(zhì)定理.利用馬氏鏈的強極限定理,得到了隱非齊次馬爾可夫模型的強大數(shù)定律.

隱馬爾科夫模型;隱非齊次馬爾可夫模型;強大數(shù)定律

1 引言

設(shè){Xn,n≥0}與{Yn,n≥0}是概率空間(?,F,P)上分別取值于S與T的隨機變量序列,其中S與T均為可列集.假設(shè){Xn,n≥0}是馬氏鏈，它不能被直接觀測到,稱為隱藏鏈.而能觀測到的是{Yn,n≥0}，稱為觀測鏈.若對?xt∈S,yt∈T,0≤t≤n,有

則稱(Xn,Yn,n≥0)為隱馬爾可夫模型.當(dāng){Xn,n≥0}是非齊次馬氏鏈時,稱(Xn,Yn,n≥0)為隱非齊次馬爾可夫模型.

隱馬爾可夫模型在近幾十年被廣泛應(yīng)用于弱相依隨機變量的建模上,主要用于發(fā)音過程,神經(jīng)生理學(xué)與生物遺傳等方面問題的研究.隱馬爾可夫模型的理論研究方面也取得了一系列的成果.Leroux[1],Genon-Catalot等[2]給出了一些隱馬爾可夫模型的熵遍歷定律, Maxwall[3]與Bickel等[4]給出了隱馬爾可夫模型在中心極限定理方面的一些性質(zhì).但隱馬爾可夫模型的理論研究并不完善,并且在實際應(yīng)用中經(jīng)常遇到隱藏鏈為非齊次馬爾可夫鏈的情形,如動態(tài)的圖象處理[5],氣候的預(yù)測[6]中均需要建立隱非齊次馬爾可夫模型,但對隱非齊次馬爾可夫模型的研究甚少.本文給出了隱馬爾可夫模型的一些性質(zhì)定理,并在狀態(tài)集都有限的情況下,給出了隱非齊次馬爾可夫模型的強大數(shù)定律.

2 模型性質(zhì)

隱馬爾可夫模型具有如下性質(zhì):

定理1設(shè){Xn,n≥0}與{Yn,n≥0}是概率空間(?,F,P)上分別取值于S與T的隨機變量序列,則下列敘述等價

(?)(Xn,Yn,n≥0)是隱馬爾可夫模型;

(??)?xt∈S,yt∈T,0≤t≤n,有

3 強大數(shù)定律

由(28)式與(19)式,即得(21)式.

4)在(21)式中設(shè)f(x,y)=f(y),由(21)式即得(22)式.

[1]Leroux B G.Maximum-likelihood estimation for hidden Markov models[J].Stochastic process Appl.,1992, 40:127-143.

[2]Genon-Catalot V,Laredo C.Leroux’s method for general hidden Markov models[J].Stochastic process Appl., 2006,116:222-243.

[3]Bickel P J,Ritov R,Ya’acov T R.Asmptotic normality of the maximum-likelihood estimator for general hidden markov models[J].Ann.Statist.,1998,26(4):1614-1635.

[4]Maxwell M,Woodroofe M.A local limit theorem for hidden Markov chains[J].Stat.Probab.Letts.,1997, 32:125-131.

[5]Beatriz L,Pilar L,Alberto L.Dynamic graphical models and nonhomogeneous hidden Markov models[J]. Statistics Probability Letters,2000,49:377-385.

[6]Bryson C,Bates S.Stochastic downscaling of numerical climate model simulations[J].Environmental Modelling Software,1998,13:325-331.

[7]楊衛(wèi)國,吳小太,王豹.一類隱馬爾可夫模型的若干極限性質(zhì)[J].江蘇大學(xué)學(xué)報:自然版,2006,27(5):467-470.

[8]王梓坤,楊向群.生滅過程與馬爾可夫鏈[M].北京:科學(xué)出版社,2005.

[9]Liu W,Yang W G.A class of strong limit theorems for the sequences of arbitrary random variables[J].Stat. Probab.Letts.,2003,64:121-131.

[10]Liu W,Yang W G.An extension of Shannon-Mcmillan theorem and some limit properties for nonhomogeneous Markov chains[J].Stochastic Process.Appl.,1996,61:129-145.

[11]汪進,楊衛(wèi)國.m重非齊次馬氏鏈的Cesaro平均收斂性[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2008,24(4):752-758.

The strong law of large numbers of nonhomogeneous hidden Markov models

WU Xiao-tai,WU Yan-lei
(Department of Mathematics,Anhui University of Technology and Science,Wuhu241000,China)

In this paper,we get some properties of hidden Markov models when the sets are limited.By the strong limited theorem of Markov chains,we obtained some strong laws of large numbers for nonhomogeneous hidden Markov model.

hidden Markov models,nonhomogeneous hidden Markov models,strong law of large numbers

O211.6

A

1008-5513(2009)03-0502-06

2007-01-22.

國家自然科學(xué)基金(10571076,10826098),安徽工程科技學(xué)院院青年基金(2007YQ025).作者簡介：吳小太(1982-),講師,研究方向：馬氏鏈.

2000MSC:60F15