Level-wise Subgeometric Convergence of the Level-increment Truncation Approximation of M/G/1-type Markov Chains

TL;DR

This paper establishes a subgeometric convergence rate for the level-increment truncation approximation of M/G/1-type Markov chains, linking the approximation error to the tail behavior of the level-increment distribution.

Contribution

It provides a new subgeometric convergence formula for the LI truncation approximation under subexponential and rank-one conditions, enhancing understanding of approximation accuracy.

Findings

Convergence rate matches the tail decay of the level-increment distribution.

Error bounds are explicitly characterized for subexponential distributions.

Results apply when the downward transition matrix is rank one.

Abstract

This paper considers the level-increment (LI) truncation approximation of M/G/1-type Markov chains. The LI truncation approximation is useful for implementing the M/G/1 paradigm, which is the framework for computing the stationary distribution of M/G/1-type Markov chains. The main result of this paper is a subgeometric convergence formula for the total variation distance between the original stationary distribution and its LI truncation approximation. Suppose that the equilibrium level-increment distribution is subexponential, and that the downward transition matrix is rank one. We then show that the convergence rate of the total variation error of the LI truncation approximation is equal to that of the tail of the equilibrium level-increment distribution and that of the tail of the original stationary distribution.