State-discretization of $V$-geometrically ergodic Markov chains and convergence to the stationary distribution

TL;DR

This paper introduces a discretization method for $V$-geometrically ergodic Markov chains that approximates the stationary distribution with computable measures, providing convergence rates and applications to autoregressive processes.

Contribution

It proposes a novel discretization scheme for $V$-geometrically ergodic Markov chains that yields computable approximations of the stationary distribution with proven convergence rates.

Findings

Convergence rate of the discretization scheme in total variation.

Application to autoregressive processes with unknown explicit stationary distributions.

Illustrations demonstrating the method's effectiveness in practical scenarios.

Abstract

Let $(X_n)_{n \in\mathbb{N}}$ be a $V$-geometrically ergodic Markov chain on a measurable space $\mathbb{X}$ with invariant probability distribution $\pi$. In this paper, we propose a discretization scheme providing a computable sequence $(\widehat\pi_k)_{k\ge 1}$ of probability measures which approximates $\pi$ as $k$ growths to infinity. The probability measure $\widehat\pi_k$ is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of $(\widehat\pi_k)_{k\ge 1}$ to $\pi$ is given. As a result, the specific case of first order autoregressive processes with linear and non-linear errors is studied. Finally, illustrations of the procedure for such autoregressive processes are provided, in particular when no explicit formula for $\pi$ is known.