Generalized invariance principles for discrete-time stochastic dynamical systems

This article, based on the typical discrete-time semi-martingale convergence theorem, establishes several generalized versions of invariance principle for describing the long-term dynamical behaviors of discrete-time stochastic dynamical systems. These principles are suitable for investigating the dynamics in autonomous or non-autonomous systems and their applicability is demonstrated via using several representative examples. Particularly for autonomous systems, the established principle renders it possible to estimate the time when an orbit, initiating outside a particular bounded set, finally enters it. Furthermore, we provide a generalized version of discrete-time semi-martingale convergence theorem, and offer a counterexample to urge attentions to some delicate conditions that must be taken into account in the use of some version of convergence theorem. (C) 2022 Elsevier Ltd. All rights reserved.