GTH Algorithm, Censored Markov Chains, and $RG$-Factorization

TL;DR

This paper reviews the GTH algorithm's numerical stability and probabilistic interpretation for Markov chains, introduces the $RG$-factorization as an LU-decomposition counterpart, and discusses censored Markov chains for infinite systems.

Contribution

It provides a comprehensive review of the GTH algorithm, links it with $RG$-factorization, and extends the concept to infinite Markov chain systems.

Findings

GTH algorithm is numerically stable and equivalent to Gaussian elimination.

$RG$-factorization serves as an LU-decomposition counterpart.

Censored Markov chains effectively approximate original chains with minimal error.

Abstract

In this paper, we provide a review on the GTH algorithm, which is a numerically stable algorithm for computing stationary probabilities of a Markov chain. Mathematically the GTH algorithm is an rearrangement of Gaussian elimination, and therefore they are mathematically equivalent. All components in the GTH algorithm can be interpreted probabilistically based on the censoring concept and each elimination in the GTH algorithm leads to a censored Markov chain. The $RG$-factorization is a counterpart to the LU-decomposition for Gaussian elimination. The censored Markov chain can also be treated as an extended version of the GTH algorithm for a system consisting of infinitely many linear equations. The censored Markov chain produces a minimal error for approximating the original chain under the $l_1$-norm.