Tail behaviour of stationary distribution for Markov chains with asymptotically zero drift

TL;DR

This paper studies the tail behavior of the stationary distribution of a positive recurrent Markov chain on the positive real line with asymptotically zero drift, showing it has a power-like heavy tail even with bounded jumps.

Contribution

It establishes the power-like asymptotic tail behavior of the invariant measure for Markov chains with asymptotically zero drift, using test functions and harmonic functions.

Findings

Invariant tail distribution is power-like heavy-tailed.

Heavy tails occur even with bounded jumps.

Analysis employs test functions and harmonic functions.

Abstract

We consider a Markov chain on $R^+$ with asymptotically zero drift and finite second moments of jumps which is positive recurrent. A power-like asymptotic behaviour of the invariant tail distribution is proven; such a heavy-tailed invariant measure happens even if the jumps of the chain are bounded. Our analysis is based on test functions technique and on construction of a harmonic function.