Error bounds for augmented truncations of discrete-time block-monotone Markov chains under subgeometric drift conditions

TL;DR

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

Contribution

It provides new upper bounds on the total variation distance for truncated Markov chains under subgeometric drift, extending to non-monotone chains dominated by monotone ones.

Findings

Upper bounds for total variation distance are established.

Results apply to chains with subgeometric drift conditions.

Application to GI/G/1-type Markov chains demonstrates practical relevance.

Abstract

This paper studies the last-column-block-augmented northwest-corner truncation (LC-block-augmented truncation, for short) of discrete-time block-monotone Markov chains under subgeometric drift conditions. The main result of this paper is to present an upper bound for the total variation distance between the stationary probability vectors of a block-monotone Markov chain and its LC-block-augmented truncation. The main result is extended to Markov chains that themselves may not be block monotone but are block-wise dominated by block-monotone Markov chains satisfying modified drift conditions. Finally, as an application of the obtained results, the GI/G/1-type Markov chain is considered.