Rates of memory loss for null recurrent Markov chains

TL;DR

This paper establishes the first estimates of memory loss rates for null recurrent Markov chains, filling a significant gap in understanding their convergence behavior.

Contribution

It provides the first theoretical bounds on the rates of memory loss specifically for null recurrent Markov chains, extending prior results from positive recurrent cases.

Findings

First estimates of memory loss rates for null recurrent chains

Demonstrates convergence behavior in total variation for these chains

Bridges a gap in the theory of Markov chain convergence

Abstract

Orey (1962) proved that for an irreducible, aperiodic, and recurrent Markov chain with transition operator $P$, the sequence $P^n (\mu - \nu)$ converges to zero in total variation for any two probability measures $\mu$ and $\nu$. In other words, all such Markov chains exhibit memory loss. While the rates of memory loss have been extensively studied for positive recurrent chains, there is a surprising lack of results for null recurrent chains. In this work, we prove the first estimates of memory loss rates in the null recurrent case.