On the strong stability of ergodic iterations

TL;DR

This paper investigates the strong stability of ergodic iterations driven by stationary processes, establishing conditions under which such processes converge regardless of initial states, with applications in autoregression, queuing, Langevin iterations, and branching processes.

Contribution

It provides new theoretical results on the strong stability of ergodic iterations without requiring assumptions on the driving sequence, extending previous work to dependent noise and multitype branching processes.

Findings

Proved strong stability under mild conditions on recursive maps.

Extended results to Langevin-type iterations with dependent noise.

Applied theory to generalized autoregression, queuing, and branching processes.

Abstract

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 between 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 generalized autoregression and queuing. Furthermore, new results are deduced for Langevin-type iterations with dependent noise and for multitype branching processes.