On geometric and algebraic transience for discrete-time Markov chains

TL;DR

This paper investigates the conditions under which discrete-time Markov chains exhibit geometric and algebraic transience, providing criteria based on drift conditions and first return times, with applications to specific chain types.

Contribution

It introduces new criteria for geometric and algebraic transience in Markov chains, extending understanding beyond ergodic cases to more general transience behaviors.

Findings

Criteria for geometric transience established

Criteria for algebraic transience established

Applications to random walk on half line and skip-free chain

Abstract

General characterizations of ergodic Markov chains have been developed in considerable detail. In this paper, we study the transience for discrete-time Markov chains on general state spaces, including the geometric transience and algebraic transience. Criteria are presented through establishing the drift condition and considering the first return time. As an application, we give explicit criteria for the random walk on the half line and the skip-free chain on nonnegative integers.