连续状态马氏链遍历性的初等证明

本文介绍了连续状态马氏链Dobrushin系数、指数强遍历性、强遍历性以及弱遍历性,在此基础上给出指数强遍历、强遍历和弱遍历之间相互等价的一个初等证明.     

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．     

渐近循环马氏链的收敛速度

引入了渐近循环马氏链的概念,它是循环马氏链概念的推广.首先研究了在强遍历的条件下,可列循环马氏链的收敛速度,作为主要结论给出了当满足不同条件时可列渐近循环马氏链的C-强遍历性,一致C-强遍历性和一致C-强遍历的收敛速度     

抽象空间中的马氏过程的强遍历性及收敛速度

<正> §1.引言Doob 在[2]中对一般状态的时齐的马氏过程的遍历性理论,作了系统的研究,得到了完满的结果.D.G.Kendall 在[8]中,J.F.C.Kingman 在[6]、[7]中,D.Vere-Jones在[5]中,对可数状态的时齐的马氏过程的遍历极限的收敛速度,作了研究,这些文章的一个共同特点是:假定对某一状态其遍历极限的收敛速度为几何速度(指数速度),证明对其它状态,其遍历极限的收敛速度亦然.然而 D.Isaacson 在[1]中,研究了可数状态时齐的马氏过程的强遍历性,而且证明了强遍历性蕴含了收敛速度是几何速度(指数速度).本文研究的是一般状态的马氏过程(时齐的或非时齐的),得到了马氏过程满足强遍历性的各种充要条件;证明了强遍历性蕴含了收敛的指数速度;找出了最佳收敛速度;并证明了在什么条件下达到最佳收敛速度.     

GI/G/1排队系统的几种收敛速度

对随机模型,可以从不同角度研究其稳定性,一种是研究其转移概率函数趋向于平稳分布的速度,即各种遍历性;另一种是研究平稳分布的尾部衰减速度.本文从这两个方面着手,找它们之间的关系,对GI/G/1排队系统,给出等待时间列几何遍历、平稳分布轻尾与服务时间分布轻尾三者等价,l-遍历、平稳分布的尾部(l-1)-阶衰减与服务时间分布的尾部l-阶衰减三者等价,最后证明出等待时间列不是强遍历.     

带移民的单生过程

本文给出了带移民单生过程唯一性、常返性、遍历、强遍历的显式判别准则和指数遍历的显式充分条件,以及0点首中时的n阶矩显式表达式.作为应用,给出了带移民生灭过程的相关性质,并且在文末讨论了几个例子的各种遍历性.     

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.     

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．     

一般形式的连续时间拟生灭过程各种遍历性判定准则

侯振挺、李晓花在 [1]已经讨论了具有某些特殊形式的拟生灭过程各种遍历性 ,我们将在此基础讨论一般形式连续时间拟生灭过程各种遍历性 ,并给出 [1]中连续时间拟生灭过程的指数遍历及多项式遍历的一个新证明 ,该证明给出了具有某些特殊条件下连续时间拟生灭过程遍历性与离散时间拟生灭过程遍历性之间关系 .     

广义Kolmogorov矩阵的遍历性

在本文中,我们证明了广义Kolmogorov矩阵所对应的Markov链的强遍历性与指数遍历性,并且给出它的最大指数遍历常数的一个下界。     

The ergodic theory of Markov chains in random environments

Summary A general formulation of the stochastic model for a Markov chain in a random environment is given, including an analysis of the dependence relations between the environmental process and the controlled Markov chain, in particular the problem of feedback. Assuming stationary environments, the ergodic theory of Markov processes is applied to give conditions for the existence of finite invariant measure (equilibrium distributions) and to obtain ergodic theorems, which provide results on convergence of products of random stochastic matrices. Coupling theory is used to obtain results on direct convergence of these products and the structure of the tail -field. State properties including classification and communication properties are discussed.     

A characterization of geometric ergodicity

Summary A homogeneous Markov chain on a countable state space can be classified as ergodic, geometrically ergodic, or strongly ergodic. Ergodicity and strong ergodicity have been characterized using the -coefficient. In this paper the -coefficient is used to characterize geometric ergodicity.     

Ergodic backward stochastic difference equations

We consider ergodic backward stochastic differential equations in a discrete time setting, where noise is generated by a finite state Markov chain. We show existence and uniqueness of solutions, along with a comparison theorem. To obtain this result, we use a Nummelin splitting argument to obtain ergodicity estimates for a discrete time Markov chain which hold uniformly under suitable perturbations of its transition matrix. We conclude with an application of this theory to a treatment of an ergodic control problem.     

Generalization bounds of ERM algorithm with V-geometrically Ergodic Markov chains

The previous results describing the generalization ability of Empirical Risk Minimization (ERM) algorithm are usually based on the assumption of independent and identically distributed (i.i.d.) samples. In this paper we go far beyond this classical framework by establishing the first exponential bound on the rate of uniform convergence of the ERM algorithm with V-geometrically ergodic Markov chain samples, as the application of the bound on the rate of uniform convergence, we also obtain the generalization bounds of the ERM algorithm with V-geometrically ergodic Markov chain samples and prove that the ERM algorithm with V-geometrically ergodic Markov chain samples is consistent. The main results obtained in this paper extend the previously known results of i.i.d. observations to the case of V-geometrically ergodic Markov chain samples.     

随机环境中的马氏链的不变测度与遍历性

本文考虑了一类特殊的随机环境的马氏链。假设随机“Doeblin”条件成立，我们证明了随机环境的马氏链的不变测度存在，且任何初始分布以指数收敛速度到些不变测度。进一步的，存在关于绕积算子遍历的不变测度。最后，我们得到了随机马氏链的强大数定律。     

Hypoellipticity and Ergodicity of the Wonham Filter as a Diffusion Process

The ergodicity problem of the Wonham filter as a diffusion process is discussed. Since the Wonham equation is degenerate, we apply an ergodic theorem of degenerate Markov diffusions to the problem. Under a certain condition, the Wonham equation satisfies Hörmander’s condition, and the Wonham filter has a continuous transition density. From these results, we obtain that the Wonham filter is uniformly ergodic as a diffusion process.     

A note on a variation of Doeblin's condition for uniform ergodicity of Markov chains

Summary We introduce a simple variation of Doeblin's condition, Condition D*, that assures the uniform ergodicity of a Markov chain. It is also shown that for non-homogeneous chains our conditions are equivalent to Dobrushin's weak ergodic coefficient.     

Geometric ergodicity for classes of homogeneous Markov chains

The paper deals with non asymptotic computable bounds for the geometric convergence rate of homogeneous ergodic Markov processes. Some sufficient conditions are stated for simultaneous geometric ergodicity of Markov chain classes. This property is applied to nonparametric estimation in ergodic diffusion processes.