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.

本文基于典型的离散时间半鞅收敛定理，建立了描述离散时间随机动力系统长期动力学行为的几个广义版本的不变性原理。这些原理适用于研究自治或非自治系统的动力学，并通过使用几个代表性示例证明了它们的适用性。特别是对于自治系统，既定原则使得可以估计从特定有界集之外开始的轨道最终进入它的时间。此外，我们提供了离散时间半鞅收敛定理的广义版本，并提供了一个反例以敦促注意在使用某些版本的收敛定理时必须考虑的一些微妙条件。