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

Given a sequence (Mk, Qk)k ≥ 1 of independent and identically distributed random vectors with nonnegative components, we consider the recursive Markov chain (Xn)n ≥ 0, defined by the random difference equation Xn = MnXn - 1 + Qn for n ≥ 1, where X0 is independent of (Mk, Qk)k ≥ 1. Criteria for the null recurrence/transience are provided in the situation where (Xn)n ≥ 0 is contractive in the sense that M1 ⋯ Mn → 0 almost surely, yet occasional large values of the Qn overcompensate the contractive behavior so that positive recurrence fails to hold. We also investigate the attractor set of (Xn)n ≥ 0 under the sole assumption that this chain is locally contractive and recurrent.