State-dependent Foster-Lyapunov criteria for subgeometric convergence of Markov chains

TL;DR

This paper introduces state-dependent Foster-Lyapunov criteria for Markov chains, providing conditions for subgeometric convergence and finite moments of return times, addressing questions about 'tame' chains.

Contribution

It presents new criteria linking subsampled drift conditions to subgeometric convergence and the existence of finite return time moments in Markov chains.

Findings

Criteria for state-dependent drift conditions are established.

Subsampled drift conditions imply finite moments for return times.

Partially answers the question on existence of 'tame' Markov chains.

Abstract

We consider a form of state-dependent drift condition for a general Markov chain, whereby the chain subsampled at some deterministic time satisfies a geometric Foster-Lyapunov condition. We present sufficient criteria for such a drift condition to exist, and use these to partially answer a question posed by Connor & Kendall (2007) concerning the existence of so-called 'tame' Markov chains. Furthermore, we show that our 'subsampled drift condition' implies the existence of finite moments for the return time to a small set.