Markov chains in random environment with applications in queueing theory and machine learning

TL;DR

This paper establishes the existence of limiting distributions and laws of large numbers for a broad class of Markov chains in random environments, with applications in queueing theory, machine learning, and autoregressive processes.

Contribution

It introduces new conditions ensuring convergence and stability of Markov chains in random environments, extending previous results to more general settings.

Findings

Proves existence of limiting distributions under drift and minorization conditions.

Establishes law of large numbers for bounded functionals of the process.

Demonstrates applications in queueing systems, machine learning, and autoregressive models.

Abstract

We prove the existence of limiting distributions for a large class of Markov chains on a general state space in a random environment. We assume suitable versions of the standard drift and minorization conditions. In particular, the system dynamics should be contractive on the average with respect to the Lyapunov function and large enough small sets should exist with large enough minorization constants. We also establish that a law of large numbers holds for bounded functionals of the process. Applications to queuing systems, to machine learning algorithms and to autoregressive processes are presented.