On $L_1$-Weak Ergodicity of nonhomogeneous discrete Markov processes and its applications

TL;DR

This paper investigates the conditions under which nonhomogeneous discrete Markov processes with general state spaces exhibit $L_1$-weak ergodicity, a weaker form of ergodicity, and applies these results to quadratic stochastic processes.

Contribution

It provides a necessary and sufficient condition for $L_1$-weak ergodicity in nonhomogeneous Markov processes and extends the results to quadratic stochastic processes.

Findings

Established a necessary and sufficient condition for $L_1$-weak ergodicity.

Applied the results to quadratic stochastic processes.

Provided concrete examples illustrating the theory.

Abstract

In the present paper we investigate the $L_1$-weak ergodicity of nonhomogeneous discrete Markov processes with general state spaces. Note that the $L_1$-weak ergodicity is weaker than well-known weak ergodicity. We provide a necessary and sufficient condition for such processes to satisfy the $L_1$-weak ergodicity. Moreover, we apply the obtained results to establish $L_1$-weak ergodicity of discrete time quadratic stochastic processes. As an application of the main result, certain concrete examples are also provided.