随机环境中马氏链的最小闭集

主要研究了随机环境中马氏链的最小闭集的一些性质,并就保守集C在何种情况下存在最小闭子集的开问题结合Foguel的L1-理论进行了讨论,得到了一些结果.     

Markov chains in random environments and random iterated function systems

We consider random iterated function systems giving rise to Markov chains in random (stationary) environments. Conditions ensuring unique ergodicity and a ``pure type' characterization of the limiting ``randomly invariant' probability measure are provided. We also give a dimension formula and an algorithm for simulating exact samples from the limiting probability measure.

     

Infinitely dimensional control Markov branching chains in random environments

First of all we introduce the concepts of infinitely dimensional control Markov branching chains in random environments (β-MBCRE) and prove the existence of such chains, then we introduce the concepts of conditional generating functionals and random Markov transition functions of such chains and investigate their branching property. Base on these concepts we calculate the moments of the β-MBCRE and obtain the main results of this paper such as extinction probabilities, polarization and proliferation rate. Finally we discuss the classification of β-MBCRE according to the different standards.     

The occurrence of sequence patterns in ergodic Markov chains

The occupation measure identity is used to derive the expected waiting time for the first occurrence of a fixed finite pattern in a sequence of observations generated by an ergodic Markov chain.     

On potentials of ergodic Markov chains

Two theorems on the existence of the potential of an ergodic Markov chain in an arbitrary phase space are proved.DeceasedTranslated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 446–449, April, 1994.This work was supported by the Ukrainian State Committee on Science and Technology.     

Recurrence and invariant measure of Markov chains in double-infinite random environments

The concepts of π -irreduciblity, recurrence and transience are introduced into the research field of Markov chains in random environments. That a π -irreducible chain must be either recurrent or transient is proved, a criterion is shown for recurrent Markov chains in double-infinite random environments, the existence of invariant measure of π -irreducible chains in double-infinite environments is discussed, and then Orey’s open-questions are partially answered.     

随机环境中马氏链状态的常返性与暂留性

给出了随机环境中马氏链状态必然是弱常返或强暂留的几个充分条件,引入了状态周期的概念,得到类似于经典马氏链状态周期的几个性质.引入了随机环境中马氏链状态的几个数字特征,给出了随机环境中马氏链状态是弱常返与强常返等价的充分条件,利用这一条件可以说明相关文献所出现的错误结论.     

Learning from uniformly ergodic Markov chains

Evaluation for generalization performance of learning algorithms has been the main thread of machine learning theoretical research. The previous bounds describing the generalization performance of the empirical risk minimization (ERM) algorithm are usually established based on independent and identically distributed (i.i.d.) samples. In this paper we go far beyond this classical framework by establishing the generalization bounds of the ERM algorithm with uniformly ergodic Markov chain (u.e.M.c.) samples. We prove the bounds on the rate of uniform convergence/relative uniform convergence of the ERM algorithm with u.e.M.c. samples, and show that the ERM algorithm with u.e.M.c. samples is consistent. The established theory underlies application of ERM type of learning algorithms.     

随机环境中马氏链状态的各种常返性与暂留性

考虑到随机环境中马氏链的状态在受到环境因素各种条件的影响下,引入了随机环境中马氏链状态的各种常返性与暂留性概念,讨论了这些常返性与暂留性的相互关系,从而说明随机环境中马氏链状态的常返性与暂留性和经典马氏链状态的常返性与暂留性有着显著的区别.     

The ergodic theorems for Markov chains with an arbitrary phase space

The ergodic theorems are formulated both is the case of the uniform ergodicity and in the absence of this latter under more general conditions than before. The conditions laid down on the transition function in the case of nonuniform ergodicity are less restrictive than these by Athreya-Ney and Nummelin. In contrast to these latter the restrictions are set not on the initial transition function, but on that of the imbedded Markov chain which is constructed by the instants of the recurrence to some fixed set. The proof is based on the spectral theory of Banach algebras.     

The tail empirical process of regularly varying functions of geometrically ergodic Markov chains

We consider a stationary regularly varying time series which can be expressed as a function of a geometrically ergodic Markov chain. We obtain practical conditions for the weak convergence of the tail array sums and feasible estimators of cluster statistics. These conditions include the so-called geometric drift or Foster–Lyapunov condition and can be easily checked for most usual time series models with a Markovian structure. We illustrate these conditions on several models and statistical applications. A counterexample is given to show a different limiting behavior when the geometric drift condition is not fulfilled.     

On the central limit theorem for geometrically ergodic Markov chains

Let X₀,X₁,... be a geometrically ergodic Markov chain with state space and stationary distribution . It is known that if h: R satisfies (|h|²⁺)< for some >0, then the normalized sums of the X_is obey a central limit theorem. Here we show, by means of a counterexample, that the condition (|h|²⁺)< cannot be weakened to only assuming a finite second moment, i.e., (h²)<.Reasearch supported by the Swedish Research Council.     

Strongly ergodic Markov chains and rates of convergence using spectral conditions

For finite Markov chains the eigenvalues of P can be used to characterize the chain and also determine the geometric rate at which Pⁿ converges to Q in case P is ergodic. For infinite Markov chains the spectrum of P plays the analogous role. It follows from Theorem 3.1 that 6Pⁿ?Q6?Cβⁿ if and only if P is strongly ergodic. The best possible rate for β is the spectral radius of P?Q which in this case is the same as sup{|λ|: λ ? σ (P), λ ≠;1}. The question of when this best rate equals δ(P) is considered for both discrete and continous time chains. Two characterizations of strong ergodicity are given using spectral properties of P? Q (Theorem 3.5) and spectral properties of a submatrix of P (Theorem 3.16).     

双无限环境中马氏链的强极限定理

在双无限环境中马氏链的过程矩满足一定的条件下,通过停时和鞅收敛定理,得到双无限环境中马氏链的一类强极限定理.     

A theorem on the Markov periodic approximation in ergodic theory

One presents a new variant of the theory of periodic approximations of dynamical systems and C^*-algebras, namely the construction for each automorphism of the Lebesgue space of a Markov tower (or adic model) of periodic automorphisms. One gives several examples.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 72–82, 1982.     

Separable solutions for Markov processes in random environments

In this paper we address the problem of efficiently deriving the steady-state distribution for a continuous time Markov chain (CTMC) S evolving in a random environment E. The process underlying E is also a CTMC. S is called Markov modulated process. Markov modulated processes have been widely studied in literature since they are applicable when an environment influences the behaviour of a system. For instance, this is the case of a wireless link, whose quality may depend on the state of some random factors such as the intensity of the noise in the environment. In this paper we study the class of Markov modulated processes which exhibits separable, product-form stationary distribution. We show that several models that have been proposed in literature can be studied applying the Extended Reversed Compound Agent Theorem (ERCAT), and also new product-forms are derived. We also address the problem of the necessity of ERCAT for product-forms and show a meaningful example of product-form not derivable via ERCAT.     

A fluctuation theory for Markov chains

A fluctuation theory for Markov chains on an ordered countable state space is developed, using ladder processes. These are shown to be Markov renewal processes. Results are given for the joint distribution of the extremum (maximum or minimum) and the first time the extremum is achieved. Also a new classification of the states of a Markov chain is suggested. Two examples are given.