关于可列非齐次马氏链的若干极限定理

设{Xn,n≥0}是一列非齐次马尔科夫链,{an,n≥0}是一列固定的非负整数序列.首先构造了一个带参数的广义似然比函数,然后利用Borel-Cantelli引理证明随机变量序列几乎处处收敛性,得到了关于可列非齐次马氏链序偶广义平均的若干极限定理,推广了已有的结果.     

Mixed ratio limit theorems for Markov processes

LetP be a conservative and ergodic Markov operator onL ₁(X, Σ,m). We give a sufficient condition for the existence of a decompositionA _f ↑X such that for 0≦f, g ∈L _∞ (A _j ) and any two probability measuresμ andν weaker thanm , whereλ is theσ-finite invariant measure (which necessarily exists). Processes recurrent in the sense of Harris are shown to have this decomposition, and an analytic proof of the convergence of is deduced for such processes. This paper is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the direction of Professor S. R. Foguel, to whom the author is grateful for his helpful advice and kind encouragement.     

Uniform limit theorems for Harris recurrent Markov chains

Summary We study uniform limit theorems for regenerative processes and get strong law of large numbers and central limit theorem of this type. Then we apply those results to Harris recurrent Markov chains based on some ideas of K. Athreya, P. Ney and E. Nummelin.     

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 ratio limit theorem for the finite nonhomogeneous Markov chains

A new simple proof concerning the structure of the tail σ-field of a nonhomogeneous Markov chain as well as a result regarding the asymptotic behaviour of the ratio between the transition probabilities and the absolute probabilities of the chain are given. It is shown that the latter result can be used to characterize the atomic sets of the tail σ-field.     

Isomorphism theorems for Markov chains

The isomorphism theorem of Dynkin is definitely an important tool to investigate the problems raised in terms of local times of Markov processes. This theorem concerns continuous time Markov processes. We give here an equivalent version for Markov chains.     

Local limit theorems for transition densities of Markov chains converging to diffusions

We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Local limit theorems for transition densities are proved. Received: 28 August 1998 / Revised version: 6 September 1999 / Published online: 14 June 2000     

Some limit theorems for Markov processes

The aim of this paper is to prove some limit theorems for Markov processes using only functional analytic methods. Some of our results were proved in [7], [8] and [5] by probabilistic methods. We prove in the Appendix a theorem on Markov processes that have no finite invariant measure. This paper is a part of the author’s Ph.D. thesis to be submitted to the Hebrew University of Jerusalem. The author wishes to express his thanks to Professor S. R. Foguel for much valuable advice and encouragement.     

Ratio limit theorems for Markov processes

Convergence of andμP ⁿ(B)/μP ⁿ(a) is established for a certain class of Markov operators,P, whereμ is a measure andB is a subset ofA. The results are proved under certain conditions onP and the setA.     

Local limit theorems on the convergence of Markov chains to diffusion processes

One proves the uniform convergence of the densities of the finite-dimensional distributions of certain families of Markov chains to the densities of the finite-dimensional distributions of a nondegenerate diffusion process.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 277–292, 1986.The author is grateful to S. A. Molchanov for numerous useful discussions.     

Central limit theorems for ergodic continuous-time Markov chains with applications to single birth processes

We obtain sufficient criteria for central limit theorems (CLTs) for ergodic continuous-time Markov chains (CTMCs). We apply the results to establish CLTs for continuous-time single birth processes. Moreover, we present an explicit expression of the time average variance constant for a single birth process whenever a CLT exists. Several examples are given to illustrate these results.     

Classical limit theorems for measure-valued Markov processes

Laws of large numbers, central limit theorems, and laws of the iterated logarithm are obtained for discrete and continuous time Markov processes whose state space is a set of measures. These results apply to each measure-valued stochastic process itself and not simply to its real-valued functionals.     

Strong approximation theorems for density dependent Markov chains

A variety of continuous parameter Markov chains arising in applied probability (e.g. epidemic and chemical reaction models) can be obtained as solutions of equations of the form

$\text{X}{}_{\mathrm{N}}\text{(t)=x}{}_0\text{+∑}\text{1}\text{N}\text{lY}{}_1\text{N ∫}{}^{\mathrm{t}}{}_0\text{f}{}_1\text{(X}{}_{\mathrm{N}}\text{(s))ds}$

where $\text{l∈}\text{Z}{}^{\mathrm{t}}$, the Y₁ are independent Poisson processes, and N is a parameter with a natural interpretation (e.g. total population size or volume of a reacting solution).The corresponding deterministic model, satisfies

$\text{X(t)=x}{}_0\text{+ ∫}{}^{\mathrm{t}}{}_0\text{∑ lf}{}_1\text{(X(s))ds}$

Under very general conditions lim_N→∞X_N(t)=X(t) a.s. The process X_N(t) is compared to the diffusion processes given by

$\text{Z}{}_{\mathrm{N}}\text{(t)=x}{}_0\text{+∑}\text{1}\text{N}\text{lB}{}_1\text{N∫}{}^{\mathrm{t}}{}_0\text{ft(Z}{}_{\mathrm{N}}\text{(s))ds}$

and

$\text{V(t)=∑ l∫}{}^{\mathrm{t}}{}_0\text{f}{}_1\text{(X(s))}\text{d}\text{W}\text{?}{}_1\text{+∫}{}^{\mathrm{t}}{}_0\text{?F(X(s))·V(s)}\text{ds.}$

Under conditions satisfied by most of the applied probability models, it is shown that X_N,Z_N and V can be constructed on the same sample space in such a way that

$\text{X}{}_{\mathrm{N}}\text{(t)=Z}{}_{\mathrm{N}}\text{(t)+O}\text{logN}\text{N}$

and

$\text{N}\text{(X}{}_{\mathrm{N}}\text{(t)?X(t))=V(t)+O}\text{log N}\text{N}$

     

Limit theorems for Markov chains of finite rank

We consider a Markov chain with a general state space, but whose behavior is governed by finite matrices. After a brief exposition of the basic properties of this chain, its convenience as a model is illustrated by three limit theorems. The ergodic theorem, the central limit theorem, and an extreme-value theorem are expressed in terms of dominant eigenvalues of finite matrices and proved by simple matrix theory.     

Central limit theorems for supercritical branching Markov processes

In this paper we establish spatial central limit theorems for a large class of supercritical branching Markov processes with general spatial-dependent branching mechanisms. These are generalizations of the spatial central limit theorems proved in [1] for branching OU processes with binary branching mechanisms. Compared with the results of [1], our central limit theorems are more satisfactory in the sense that the normal random variables in our theorems are non-degenerate.