On the asymptotic events of a Markov chain

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.

Keywords

• tail σ-field
• invariant σ-field
• atomic set
• completely nonatomic set
• harmonic function
• space-time harmonic function
• martingale
• Martin boundary
• Chacon-Ornstein ergodic theorem
• 0−2 Laws.