On Convergence of a Truncation Scheme for Approximating Stationary Distributions of Continuous State Space Markov Chains and Processes

TL;DR

This paper proves that truncating unbounded state space Markov chains and processes yields stationary distributions that converge to the original distribution, using regeneration principles, with broad applicability including countable state spaces.

Contribution

It establishes general convergence results for stationary distributions of truncated Markov chains and processes, extending existing theories to broader classes including countable state spaces.

Findings

Stationary distributions of truncated chains converge in total variation norm.

Results apply to positive Harris recurrent chains and certain Harris recurrent processes.

Extension to non-explosive Markov jump processes on countable state spaces.

Abstract

In the analysis of Markov chains and processes, it is sometimes convenient to replace an unbounded state space with a "truncated" bounded state space. When such a replacement is made, one often wants to know whether the equilibrium behavior of the truncated chain or process is close to that of the untruncated system. For example, such questions arise naturally when considering numerical methods for computing stationary distributions on unbounded state space. In this paper, we use the principle of "regeneration" to show that the stationary distributions of "fixed state" truncations converge in great generality (in total variation norm) to the stationary distribution of the untruncated limit, when the untruncated chain is positive Harris recurrent. Even in countable state space, our theory extends known results by showing that the augmentation can correspond to an $r$-regular measure. In addition, we extend our theory to cover an important subclass of Harris recurrent Markov processes that include non-explosive Markov jump processes on countable state space.