随机环境中多维分枝链的增长率及灭绝概率

本文在随机环境中马氏链的框架下研究随机环境中多维分枝链(简记MBCRE)的极限性质,获得了MBCRE的母函数的一些性质,利用这些性质和随机矩阵乘积的弱收敛性证明了MBCRE中的条件均方收敛性与a．s．收敛性以及灭绝概率的估算等．这些结果是对Athrey与Karlin(1972)和Cohn(1989)的极限定理的推广．     

The branching chain with drift in space-time random environment (I): Model,Markov property,moments

There are three parts in this article. In Section 1, we establish the model of branching chain with drift in space-time random environment (BCDSTRE), i.e., the coupling of branching chain and random walk. In Section 2, we prove that any BCDSTRE must be a Markov chain in time random environment when we consider the distribution of the particles in space as a random element. In Section 3, we calculate the first-order moments and the second-order moments of BCDSTRE.     

随机环境中分枝q-过程

引进了机环境中一致马氏过程,随机分枝q-矩阵和随机环境中分枝q-过程.给了随机环境中分枝q-过程存在性和唯一性的充分条件.最后证明了任意随机分枝转移密度矩阵都是零流入的.     

ON MARKOV CHAINS IN SPACE-TIME RANDOM ENVIRONMENTS

In Section 1, the authors establish the models of two kinds of Markov chains in space-time random environments (MCSTRE and MCSTRE(+)) with abstract state space. In Section 2, the authors construct a MCSTRE and a MCSTRE(+) by an initial distribution Φ and a random Markov kernel (RMK) p(γ). In Section 3, the authors es-tablish several equivalence theorems on MCSTRE and MCSTRE(+). Finally, the authors give two very important examples of MCMSTRE, the random walk in spce-time random environment and the Markov br...     

THE BRANCHING CHAIN WITH DRIFT IN SPACE-TIME RANDOM ENVIRONMENT （I）： MODEL,MARKOV PROPERTY,MOMENTS

There are three parts in this article. In Section 1, we establish the model of branching chain with drift in space-time random environment （BCDSTRE）, i.e., the coupling of branching chain and random walk. In Section 2, we prove that any BCDSTRE must be a Markov chain in time random environment when we consider the distribution of the particles in space as a random element. In Section 3, we calculate the first-order moments and the second-order moments of BCDSTRE.     

THE EXISTENCE AND MOMENTS OF CANONICAL BRANCHING CHAIN IN RANDOM ENVIRONMENT

The concepts of branching chain in random environmnet and canonical branch-ing chain in random environment are introduced. Moreover the existence of these chains is proved. Finally the exact formulas of mathematical expectation and variance of branching chain in random environment are also given.     

Second order limit theorems for the Markov branching process in random environments

In a Markov branching process with random environments, limiting fluctuations of the population size arise from the changing environment, which causes random variation of the ‘deterministic’ population prediction, and from the stochastic wobble around this ‘deterministic’ mean, which is apparent in the ordinary Markov branching process. If the random environment is generated by a suitable stationary process, the first variation typically swamps the second kind. In this paper, environmental processes are considered which, in contrast, lead to sampling and environmental fluctuation of comparable magnitude. The method makes little use either of stationarity or of the branching property, and is amenable to some generalization away from the Markov branching process.     

关于随机环境中的马尔可夫过程的简介——纪念李国平院士吴新谋教授诞辰100周年

该文系统地介绍随机环境中的马尔可夫过程. 共4章, 第一章介绍依时的随机环境中的马尔可夫链(MCTRE), 包括MCTRE的存在性及等价描述; 状态分类; 遍历理论及不变测度; p-θ 链的中心极限定理和不变原理. 第二章介绍依时的随机环境中的马尔可夫过程(MPTRE), 包括MPTRE的基本概念; 随机环境中的q -过程存在唯一性; 时齐的q -过程;MPTRE的构造及等价性定理.第三章介绍依时的随机环境中的分枝链(MBCRE), 包括有限维的和无穷维的MBCRE的模型和基本概念; 它们的灭绝概念;两极分化; 增殖率等.第四章介绍依时依空的随机环境中的马尔可夫链(MCSTRE), 包括MCSTRE的基本概念、构造; 依时依空的随机环境中的随机徘徊(RWSTRE)的中心极限定理、不变原理.     

随机环境中具有随机控制函数的受控分枝过程

对随机环境中具有随机控制函数的受控分枝过程进行了更为详尽的概率描述和直观解释;证明了此过程是时齐马氏链和随机环境中的马氏链,并对其概率母函数及矩量进行了讨论。     

依状态独立的随机环境中的马氏链

本主要讨论了依状态独立的随机环境中的马氏链，并严格地证明了依状态独立的随机环境中的马氏链，如果在环境不退化的一般情形下，不是时齐马氏链，而环境退化必然是马氏链这个结论。     

THE DECOMPOSITION OF STATE SPACE FOR MARKOV CHAIN IN RANDOM ENVIRONMENT

This paper is a continuation of [8] and [9]. The author obtains the decomposition of state space X of an Markov chain in random environment by making use of the results in [8] and [9], gives three examples, random walk in random environment, renewal process in random environment and queue process in random environment, and obtains the decompositions of the state spaces of these three special examples.     

RENEWAL THEOREM FOR （L, 1）-RANDOM WALK IN RANDOM ENVIRONMENT

We consider a random walk on Z in random environment with possible jumps {-L,…, -1, 1}, in the case that the environment {ωi ： i ∈ Z} are i.i.d.. We establish the renewal theorem for the Markov chain of ＂the environment viewed from the particle＂ in both annealed probability and quenched probability, which generalize partially the results of Kesten （1977） and Lalley （1986） for the nearest random walk in random environment on Z, respectively. Our method is based on （L, 1）-RWRE formulated in Hong and Wang the intrinsic branching structure within the （2013）.     

Population Size Dependent Generalized Multitype Branching Processes

Abstract

In this paper, we introduce the population size dependent generalized multitype branching process. This is a Markovian model that allows us to study homogeneous multitype branching processes in a unified way. The basic properties for this model, transitions between its states, as well as the existence of a stationary limiting distribution, are investigated. Finally, we apply the obtained results to a new controlled multitype branching process.     

The Construction of Denumerable q-Processes in Random Environments-The Existence and Uniqueness

The concepts of Markov process in random environment, q-matrix in random environment, and q-process in random environment are introduced. The minimal q-process in random environment is constructed and the necessary and sufficient conditions for the uniqueness of q-process in random environment are given.     

On total progeny of multitype Galton-Watson process and the first passage time of random walk on lattice

In this paper,we form a method to calculate the probability generating function of the total progeny of multitype branching process.As examples,we calculate probability generating function of the total progeny of the multitype branching processes within random walk which could stay at its position and(2-1) random walk.Consequently,we could give the probability generating functions and the distributions of the first passage time of corresponding random walks.Especially,for recurrent random walk which could stay at its position with probability 0 r 1,we show that the tail probability of the first passage time decays as 2/(π(1-r)~(1/2)) n~(1/1)= when n →∞.     

随机环境中带移民的上临界分枝过程的Berry-Esseen界

考虑独立同分布的随机环境中带移民的上临界分枝过程(Zn).应用(Zn)与随机环境中不带移民分枝过程的联系,以及与相应随机游动的联系,在一些适当的矩条件下,本文证明关于log Zn的中心极限定理的Berry-Esseen界.     

从p—m链到随机环境中的马氏链

第一节引进了p一m链的概念,并用之构造了与它相应的随机环境中的马氏链和绕积马氏链、第二节引进了一系列与随机环境中的马氏链相关的概率特性函数,并得到了这些函数之间的一系列关系.这些结果是经典马氏链的相应结果的一般化,它们在随机环境中的马氏链的极限理论的研究中是很有用的。     

A note on multitype branching process with bounded immigration in random environment

In this paper, we study the total number of progeny, W, before regenerating of multitype branching process with immigration in random environment. We show that the tail probability of |W| is of order t-κ as t→∞, with κ some constant. As an application, we prove a stable law for (L-1) random walk in random environment, generalizing the stable law for the nearest random walk in random environment (see "Kesten, Kozlov, Spitzer: A limit law for random walk in a random environment. Compositio Math., 30, 145-168 (1975)").     

随机环境中伴有移民两性分枝过程的极限性质

本文在独立同分布的随机环境下,建立带有移民的两性分枝过程{Zn}n≥0,且移民人口数依赖当前人口数,证得{Zn}n≥0和{(Fn,Mn)}n≥1是随机环境中的马氏链,并得到第n代每个配对单元平均增长率{rk}k≥0的极限性质,从而推广了经典两性分枝过程的相关理论.     

THE ERGODICITY FOR BI-IMMIGRATION BIRTH AND DEATH PROCESSES IN RANDOM ENVIRONMENT

The concepts of bi-immigration birth and death density matrix in random environment and bi-immigration birth and death process in random environment are introduced. For any bi-immigration birth and death matrix in random environment Q（θ） with birth rate λ 〈 death rate μ, the following results are proved, （1） there is an unique q-process in random environment, P^-（θ＊（0）;t） = （p^-（θ^＊（0）;t,i,j）,i,j ≥ 0）, which is ergodic, that is, lim t→∞（θ^＊（0）;t,i,j） = π^-（θ^＊（0）;j） ≥0 does not depend on i ≥ 0 and ∑j≥0π （θ＊（0）;j） = 1, （2） there is a bi-immigration birth and death process in random enjvironment （X^＊ = {X^＊,t ≥ 0},ε^＊ = {εt,t ∈ （-∞, ∞）}） with random transition matrix P^-（θ^＊ （0）;t） such that X^＊ is a strictly stationary process.