Strong transience for one-dimensional Markov chains with asymptotically zero drifts

TL;DR

This paper investigates the transience properties of one-dimensional Markov chains with near-critical, asymptotically zero drifts, providing quantitative measures of transience through moments of return and exit times using advanced probabilistic techniques.

Contribution

It introduces a method to quantify transience in Lamperti-type Markov chains by analyzing moments of return times, employing a Doob $h$-transform and deriving asymptotic expansions.

Findings

Return probability is regularly varying with starting point.

Conditioned process remains of Lamperti type with transformed parameters.

Quantitative transience measures via moments of return and exit times.

Abstract

For near-critical, transient Markov chains on the non-negative integers in the Lamperti regime, where the mean drift at $x$ decays as $1/x$ as $x \to \infty$, we quantify degree of transience via existence of moments for conditional return times and for last exit times, assuming increments are uniformly bounded. Our proof uses a Doob $h$-transform, for the transient process conditioned to return, and we show that the conditioned process is also of Lamperti type with appropriately transformed parameters. To do so, we obtain an asymptotic expansion for the ratio of two return probabilities, evaluated at two nearby starting points; a consequence of this is that the return probability for the transient Lamperti process is a regularly-varying function of the starting point.