Ergodic and stability theorems for a class of stochastic equations and their applications

Let $\{\tau_j,\infty<j<\infty\}$ be a vector valued stationary metrically transitive sequence and let the sequence $w_n$ (also vector valued) be defined by relations $w_{n+1}=f(w_n,\tau_n)$, $n\ge 1$. We study the conditions under which the sequence $\{w_{n+k}\colon k\ge 0\}$ converges to some stationary sequence $\{w^k\colon k\ge 0\}$ as $n\to\infty$, and the conditions, under which the latter will be stable when the variations of the governing sequence $\{\tau_j\}$ are small. Applications to many-channel queueing systems are considered.