We revisit processes generated by iterated random functions driven by a stationary and ergodic sequence. Such a process is called strongly stable if a random initialization exists, for which the process is stationary and ergodic, and for any other initialization, the difference of the two processes converges to zero almost surely. Under some mild conditions on the corresponding recursive map, without any condition on the driving sequence, we show the strong stability of iterations. Several applications are surveyed such as stochastic approximation and queuing. Furthermore, new results are deduced for Langevin-type iterations with dependent noise and for multitype branching processes.

翻译：本文重新考察了由平稳遍历序列驱动的迭代随机函数生成的过程。若存在随机初始值使得该过程具有平稳遍历性，且对于任意其他初始值，两过程的差值几乎必然收敛于零，则称该过程具有强稳定性。在相应递归映射的温和条件下，无需对驱动序列施加任何约束，我们证明了迭代过程的强稳定性。本文综述了随机逼近和排队论等多个应用方向。此外，我们还推导出依赖噪声的Langevin型迭代过程及多类型分支过程的全新结论。