Recurrence and transience of random difference equations in the critical case

TL;DR

This paper investigates the long-term behavior of a specific class of random difference equations in the critical oscillating case, establishing conditions for recurrence and transience of the associated Markov chain.

Contribution

It provides new criteria for null-recurrence and transience in the critical case, extending previous techniques to this nuanced scenario.

Findings

Conditions for null-recurrence established

Criteria for transience derived

Extension of techniques from prior work

Abstract

For i.i.d. random vectors $(M_{1},Q_{1}),(M_{2},Q_{2}),\ldots$ such that $M>0$ a.s., $Q\geq 0$ a.s. and $\mathbb{P}(Q=0)<1$, the random difference equation $X_{n}=M_{n}X_{n-1}+Q_{n}$, $n=1,2,\ldots$, is studied in the critical case when the random walk with increments $\log M_{1},\log M_{2}$ is oscillating. We provide conditions for the null-recurrence and transience of the Markov chain $(X_{n})_{n\ge 0}$ by inter alia drawing on techniques developed in the related article Alsmeyer et al (2017) for another case exhibiting the null-recurrence/transience dichotomy.