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：我们重新审视了由迭代随机函数生成、由静止和遍历序列驱动的过程。如果存在一种随机初始化，使得该过程是静止和遍历的，且对于任何其他初始化，这两个过程的差异几乎一定会收敛于零，则称这样的过程是强稳定的。在对应的递归映射存在一些温和的条件下，无需在驱动序列上做任何假设，我们展示了迭代的强稳定性。许多应用也得到了研究，如随机逼近和排队。此外，新的结果也被推导出来，例如依赖噪声的Langevin类型迭代和多类型分支过程。