On limit periodicity of discrete time stochastic processes

TL;DR

This paper studies discrete-time stochastic systems with periodic coefficients, establishing conditions under which solutions converge to a periodic process, including almost sure convergence, thus advancing understanding of stochastic periodicity.

Contribution

It provides new sufficient conditions for the existence and convergence of periodic solutions in stochastic difference equations with periodic coefficients.

Findings

Conditions for existence of periodic solutions

Almost sure convergence to periodic processes

Convergence criteria for stochastic difference equations

Abstract

We consider a discrete time dynamic system described by a difference equation with periodic coefficients and with additive stochastic noise. We investigate the possibility of the periodicity for the solution. In particular, we found sufficient conditions for existence of a periodic process such that the solution converges to it, including almost surely convergence.