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 tailσ-field and the invariantσ-field of both homogeneous and nonhomogeneous Markov chains as representations for asymptotic events, descriptions of completely nonatomic and atomic sets and global characterizations of asymptoticσ-fields. It is shown that the Martin boundary theory can provide a unified approach to the asymptoticσ-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