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.
Categories: math.PR