Rates of memory loss for null recurrent Markov chains
Ilya Chevyrev, Alexey Korepanov
TL;DR
This work extends the theory of memory loss from positive-recurrent Markov chains to the null-recurrent setting by developing three complementary approaches that apply to general Harris chains. It introduces a model renewal framework with renewal probabilities $u_n$ and tail weights $z_n$, deriving both upper and lower bounds on memory loss in terms of tail behavior and renewal increments. Through Ornstein’s coupling, Harris atoms via splitting, and renewal-theoretic analysis, the authors obtain concrete rates such as $\|P^{n+\ell}\delta_0-P^n\delta_0\|_{TV} \lesssim \ell^{2/3}\rho(n)^{1/3}$ for tails with index $-\alpha\in(-3/2,0)$ and, under DFR or regularly varying tails, explicit and sometimes sharp bounds like $\lesssim \ell^{\alpha}/n^{\alpha}$ or $\lesssim \ell^{2\alpha-1}/n^{2\alpha-1}$. These results yield the first quantitative memory-loss rates for null-recurrent Markov chains and offer a bridge to broader Harris-chain contexts and potential ergodic-theory applications via renewal-type representations.
Abstract
Orey (1962) proved that for an irreducible, aperiodic, and recurrent Markov chain with transition operator $P$, the sequence $P^n (μ- ν)$ converges to zero in total variation for any two probability measures $μ$ and $ν$. 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.