非齐次马尔科夫链的转移函数的分析性质

§1.引言到目前为止,关于非齐次马尔科夫过程的研究还不多,特别,关于它的转移函数和样本函数的分析性质的研究,就更少了。但象齐次马尔科夫过程(以后简称马氏过程)一样,研究转移函数的分析性质在整个马氏过程理论的研究中起着相当基本的和重要的作用。本文以区间函数为工具,研究了具连续参量和可数状态空间的非齐次马氏链的转移函数的分析性质,如连续性,可积性,可导性等。文中的主要结果写成12个定理。除定理4外,其余都是新的。在齐次的特殊情形,可以导出[1](Ⅱ§§1—3)中的主要结果;在有限状     

一类特殊非齐次树上马氏链的若干强大数定律

首先给出了一类特殊非齐次树上可数状态马氏链的局部收敛定理,作为推论,得到了此类树上可数状态马氏链关于状态与状态序偶出现频率的若干极限性质,最后得到了这类特殊非齐次树上有限状态马氏链关于状态与状态序偶出现频率的强大数定律.     

一类非齐次马氏链的绝对平均强遍历性

引用马氏链绝对平均强遍历的概念,首先给出齐次马氏链绝对平均强遍历与强遍历的等价性,其次通过引进另一个强遍历的非齐次马氏链,给出一个非齐次马氏链绝对平均强遍历的充分条件.     

关于非刘次马氏链的绝对平均强遍历性及若干应用

本文引进非齐次马氏链绝对平均强遍历我概念并得到非齐次马氏链满足这种强遍历的一个充分条件，最后给出绝对平均强遍历性在马氏决策过程和信息论听应用。     

非齐次马尔科夫链遍历性的一些结果

本文利用两个非齐次马尔科夫链的转移矩阵列的比较，讨论了两个链启遍历性的关系，得到一个非齐次马尔科夫链是强遍历的一些充分条件．本文还分析了非齐次马尔科夫链的一致强、弱遍历性的关系，得到一个非齐次马尔科夫链是一致强遍历的一些充分条件．     

可列非齐次马氏链的强极限性质

本文主要研究可列状态非齐次马氏链的强极限性质.文中在Cesaro收敛条件下证明了非齐次马氏链的N+1元函数的一系列强极限性质,推广了已有的关于二元随机变量函数的类似结果.最后,作为推论给出了齐次马氏链的N+1元函数的强极限定理.     

二重非齐次马氏链的遍历性及其应用

给出二重非齐次马氏链的强遍历性,绝对平均强遍历性,Cesaro平均收敛的概念．利用二维马氏链的遍历性和C-K方程,建立了二维马氏链与二重非齐次马氏链遍历性的关系．并讨论了齐次二重马氏链绝对平均强遍历与强遍历的等价性．最后给出Cesaro平均收敛在马氏决策过程和信息论中应用．     

任意树指标随机过程关于树指标非齐次马氏链随机转移概率调和平均的强偏差定理

本文给出任意树指标随机过程关于树指标非齐次马氏链随机转移概率调和平均的一个强偏差定理,并作为推论得到树指标非齐次马氏链随机转移概率调和平均的强极限定理.     

关于连续状态非齐次马氏链的绝对平均强遍历性

引进连续状态非齐次马氏链绝对平均强遍历的概念,研究连续状态非齐次马氏链满足这种强遍历的一个充分条件,并给出绝对平均强遍历性在马氏决策过程中的应用.     

关于树指标非齐次马氏链的广义熵遍历定理

主要研究了树指标非齐次马氏链的广义熵遍历定理.首先证明了树指标非齐次马氏链上的二元函数延迟平均的强极限定理.然后得到了树指标非齐次马氏链上状态出现延迟频率的强大数定律,以及树指标非齐次马氏链的广义熵遍历定理.作为推论,推广了一些已有结果.同时,证明了局部有限无穷树树指标有限状态随机过程广义熵密度的一致可积性.     

An Ergodic Decomposition Defined by Regular Jointly Measurable Markov Semigroups on Polish Spaces

For a regular jointly measurable Markov semigroup on the space of finite Borel measures on a Polish space we give a Yosida-type decomposition of the state space, which yields a parametrisation of the ergodic probability measures associated to this semigroup in terms of subsets of the state space. In this way we extend results by Costa and Dufour (J. Appl. Probab. 43:767?C781, 2006). As a consequence we obtain an integral decomposition of every invariant probability measure in terms of the ergodic probability measures. Our approach is completely centered around the reduction to and relationship with the case of a single regular Markov operator associated to the Markov semigroup, the resolvent operator, which enables us to fully exploit results in that situation (Worm and Hille in Ergod. Theory Dyn. Syst. 31(2):571?C597, 2011).     

Equivalence of the Local Markov Inequality and a Kolmogorov Type Inequality in the Complex Plane

We prove that a compact subset of the complex plane satisfies a local Markov inequality if and only if it satisfies a Kolmogorov type inequality. This result generalizes a theorem established by Bos and Milman in the real case. We also show that every set satisfying the local Markov inequality is a sum of Cantor type sets which are regular in the sense of the potential theory.     

Markov selection and W-strong Feller for 3D stochastic primitive equations

This paper studies some analytical properties of weak solutions of 3D stochastic primitive equations with periodic boundary conditions. The martingale problem associated to this model is shown to have a family of solutions satisfying the Markov property, which is achieved by means of an abstract selection principle. The Markov property is crucial to extend the regularity of the transition semigroup from small times to arbitrary times. Thus, under a regular additive noise, every Markov solution is shown to have a property of continuous dependence on initial conditions, which follows from employing the weak-strong uniqueness principle and the Bismut-Elworthy-Li formula.     

A χ2 goodness-of-fit test for Markov renewal processesgoodness-of-fit test for Markov renewal processes

A Markov Renewal Process (M.R.P.) is a process similar to a Markov chain, except that the time required to move from one state to another is not fixed, but is a random variable whose distribution may depend on the two states between which the transition is made. For an M.R.P. ofm (<∞) states we derive a goodness-of-fit test for a hypothetical matrix of transition probabilities. This test is similar to the test Bartlett has derived for Markov chains. We calculate the first two moments of the test statistic and modify it to fit the moments of a standard χ². Finally, we illustrate the above procedure numeerically for a particular case of a two-state M.R.P. Dwight B. Brock is mathematical statistican, Office of Statistical Methods, National Center for Health Statistics, Rockville, Maryland. A. M. Kshisagar is Associate Professor, Department of Statistics, Southern Methodist University. This research was partially supported by Office of Naval Research Contract No. N000 14-68-A-0515, and by NIH Training Grant GM-951, both with Southern Methodist University. This article is partially based on Dwight B. Brock's Ph.D. dissertation accepted by Southern Methodist University.     

Quasi-stationarity and quasi-ergodicity of general Markov processes

In this paper, we study the quasi-stationarity and quasi-ergodicity of general Markov processes. We show, among other things, that if X is a standard Markov process admitting a dual with respect to a finite measure m and if X admits a strictly positive continuous transition density p(t, x, y) (with respect to m) which is bounded in (x, y) for every t > 0, then X has a unique quasi-stationary distribution and a unique quasi-ergodic distribution. We also present several classes of Markov processes satisfying the above conditions.     

On One Property of a Regular Markov Chain

We prove that if a certain row of the transition probability matrix of a regular Markov chain is subtracted from the other rows of this matrix and then this row and the corresponding column are deleted, then the spectral radius of the matrix thus obtained is less than 1. We use this property of a regular Markov chain for the construction of an iterative process for the solution of the Howard system of equations, which appears in the course of investigation of controlled Markov chains with single ergodic class and, possibly, transient states.     

带灾难的非线性马尔可夫分枝过程

本文在文献[6]的基础上，集中考虑一类带灾难的非线性马尔可夫分枝过程的基本问题-唯一性，正则性和灭绝性。文章首先给出其Q-过程唯一性的证明，然后得出该畔程的正则性与[3]非线性马尔币夫分枝过程一样，最后，我们给出该Q-过程以概1l灭绝的充要条件是Q-过程正则。     

Some Markov processes with Brownian exit distributions

Summary LetD be a bounded domain inR ^d with regular boundary. LetX=(X_t, P^x) be a standard Markov process inD with continuous paths up to its lifetime. IfX satisfies some weak conditions, then it is possible to add a non-local part to its generator, and construct the corresponding standard Markov process inD with Brownian exit distributions fromD.This work was done while the author was an Alexander von Humboldt fellow at the Universität des Saarlandes in Saarbrücken, Germany     

一个广义Markov不等式（英）

This paper gives a generalized Markov inequality dx for every polynomial P of degree at most n provided that f′ is con tinuous and strictly increasing on [0,∞ ), where ‖?‖ denotes the uniform and T_n,stands for the n-th Chebyshev polynomial of the first kind.     

Shift ergodicity for stationary Markov processes

In this paper shift ergodicity and related topics are studied for certain stationary processes. We first present a simple proof of the conclusion that every stationary Markov process is a generalized convex combination of stationary ergodic Markov processes. A direct consequence is that a stationary distribution of a Markov process is extremal if and only if the corresponding stationary Markov process is time ergodic and every stationary distribution is a generalized convex combination of such extremal ones. We then consider space ergodicity for spin flip particle systems. We prove space shift ergodicity and mixing for certain extremal invariant measures for a class of spin systems, in which most of the typical models, such as the Voter Models and the Contact Models, are included. As a consequence of these results we see that for such systems, under each of those extremal invariant measures, the space and time means of an observable coincide, an important phenomenon in statistical physics. Our results provide partial answers to certain interesting problems in spin systems.