Null-recurrence and transience of random difference equations in the contractive case

TL;DR

This paper investigates the long-term behavior of a recursive Markov chain defined by a random difference equation, providing criteria for null-recurrence and transience in the contractive case, and analyzing its attractor set.

Contribution

It offers new criteria for null-recurrence and transience of the chain under contractive conditions with occasional large jumps, and studies the attractor set under local contractivity.

Findings

Criteria for null-recurrence and transience are established.

Large values of Q can prevent positive recurrence despite contractivity.

The attractor set is characterized under local contractivity and recurrence.

Abstract

Given a sequence $(M_{k}, Q_{k})_{k\ge 1}$ of independent, identically distributed ran\-dom vectors with nonnegative components, we consider the recursive Markov chain $(X_{n})_{n\ge 0}$, defined by the random difference equation $X_{n}=M_{n}X_{n-1}+Q_{n}$ for $n\ge 1$, where $X_{0}$ is independent of $(M_{k}, Q_{k})_{k\ge 1}$. Criteria for the null-recurrence/transience are provided in the situation where $(X_{n})_{n\ge 0}$ is contractive in the sense that $M_{1}\cdot\ldots\cdot M_{n}\to 0$ a.s., yet occasional large values of the $Q_{n}$ overcompensate the contractive behavior so that positive recurrence fails to hold. We also investigate the attractor set of $(X_{n})_{n\ge 0}$ under the sole assumption that this chain is locally contractive and recurrent.