Erogodic (Limit) Theorems for Non-Stationary Discrete Time Markov Chains

Abstract

In this paper we study the Ergodocity of non-stationary discrete

time Markov chains. We prove that given a sequence of Ergodic Markov

chains, then the limit of the combination of the elements of this sequence is

again Ergodic (under additional condition if the state space is infinite). We

also prove that the limit of an arbitrary sequence of Markov chains is weak

Ergodic if it satisfies some condition. Under the same condition, the limit of

the combination of doubly stochastic sequence of Markov chains is Ergodic.

Keywords: Markov Chain, Stochastic, Doubly Stochastic, Irreducible,

Aperiodic, Persistent, Transient, Ergodic, Transition Matrix, Ergodic

Theorem.

The paper is organized in the following way. In the first four sections we give

a general review of the theory of Markov chains: definitions, classifications

of the chains and main theorems. In section 5 we introduce the concept of

non-stationary Markov chains. In section 6 we give some examples of nonstationary

Markov chains. In section 7 we give some limit theorems for nonstationary

Markov chains which is our main result. In section 8 we give some

remarks.