Error Bounds for Augmented Truncations of Discrete-Time Block-Monotone Markov Chains under Geometric Drift Conditions

TL;DR

This paper derives error bounds for the approximation of stationary distributions of discrete-time block-monotone Markov chains using augmented truncations, under geometric drift conditions, with applications to GI/G/1-type chains.

Contribution

It provides new total variation distance bounds for augmented truncations of block-monotone Markov chains, including non-monotone chains dominated by monotone ones.

Findings

Established bounds for stationary distribution approximation errors.

Extended results to non-monotone chains dominated by monotone chains.

Applied bounds to GI/G/1-type Markov chains.

Abstract

This paper studies the augmented truncation of discrete-time block-monotone Markov chains under geometric drift conditions. We first present a bound for the total variation distance between the stationary distributions of an original Markov chain and its augmented truncation. We also obtain such error bounds for more general cases where an original Markov chain itself may not be block-monotone but is block-wise dominated by a block-monotone Markov chain. Finally we discuss the application of our results to GI/G/1-type Markov chains.