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：我们重新审视由平稳遍历序列驱动的迭代随机函数生成的过程。如果存在一个随机初始化使得该过程是平稳遍历的，并且对于任何其他初始化，两个过程的差几乎必然收敛到零，则称该过程为强稳定的。在相应递归映射的某些温和条件下，且不对驱动序列施加任何条件，我们证明了迭代的强稳定性。本文综述了若干应用，如随机逼近和排队论。此外，还推导了具有相关噪声的朗之万型迭代以及多类型分支过程的新结论。