On the asymptotic events of a Markov chain

Harry Cohn

doi:10.1155/s0161171279000430

Abstract

In this paper we investigate some structure properties of the tail σ‐field and the invariant σ‐field of both homogeneous and nonhomogeneous Markov chains as representations for asymptotic events, descriptions of completely nonatomic and atomic sets and global characterizations of asymptotic σ‐fields. It is shown that the Martin boundary theory can provide a unified approach to the asymptotic σ‐fields theory.

Highlights

In this paper we investigate some structure properties of the tail o-field and the invariant o-field of both homogeneous and nonhomogeneous Markov chains as representations for asymptotic events, descriptions of completely nonatomic and atomic sets and global characterizations of asymptotic o-flelds
The first result on the asymptotic events of a sequence of random variables was the 0-i law given by Kolmogorov in 1933 [28]
Etc. to obtain important properties of some variables derived from sequences of independent random variables

Summary

Introduction

In this paper we investigate some structure properties of the tail o-field and the invariant o-field of both homogeneous and nonhomogeneous Markov chains as representations for asymptotic events, descriptions of completely nonatomic and atomic sets and global characterizations of asymptotic o-flelds. I) }) 0 Since n+l(Hn-Hn+l) 0 implies n+k(Hn-Hn+l) 0 for any k >0 we can see that if a chain is improperly homogeneous, i.e. if P(lim sup{Xn e (Hn Hn+l) }) > 0 the temporary homogeneity of its transition probabilities is of no use for the sequence of sets Theorem i has a parallel result for space-time harmonic functions and tail o-fields, expressed by the following

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Results

Conclusion