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

Gerold Alsmeyer; Dariusz Buraczewski; Alexander Iksanov

paper

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

arXiv:1612.02148

Abstract

Given a sequence $(M_{k}, Q_{k})_{k \geq 1}$ of independent, identically distributed ran\-dom vectors with nonnegative components, we consider the recursive Markov chain $(X_{n})_{n \geq 0}$ , defined by the random difference equation $X_{n} = M_{n} X_{n - 1} + Q_{n}$ for $n \geq 1$ , where $X_{0}$ is independent of $(M_{k}, Q_{k})_{k \geq 1}$ . Criteria for the null-recurrence/transience are provided in the situation where $(X_{n})_{n \geq 0}$ is contractive in the sense that $M_{1} \cdot \dots \cdot M_{n} \to 0$ 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 $(X_{n})_{n \geq 0}$ under the sole assumption that this chain is locally contractive and recurrent.

submitted for publication, 24 pages