Optimal bounds for bit-sizes of stationary distributions in finite Markov chains

TL;DR

This paper establishes optimal bounds on the denominators of stationary distributions in finite Markov chains, improving previous results by replacing the Hadamard inequality with the Markov chain tree formula.

Contribution

It introduces a novel method using the Markov chain tree formula to derive sharper bounds on stationary distribution denominators, enhancing analysis of stochastic algorithms.

Findings

Derived optimal bounds for stationary distribution denominators.

Extended bounds to absorption probabilities, gains, and bias vectors.

Improved complexity estimates for stochastic mean payoff game algorithms.

Abstract

An irreducible stochastic matrix with rational entries has a stationary distribution given by a vector of rational numbers. We give an upper bound on the lowest common denominator of the entries of this vector. Bounds of this kind are used to study the complexity of algorithms for solving stochastic mean payoff games. They are usually derived using the Hadamard inequality, but this leads to suboptimal results. We replace the Hadamard inequality with the Markov chain tree formula in order to obtain optimal bounds. We also adapt our approach to obtain bounds on the absorption probabilities of finite Markov chains and on the gains and bias vectors of Markov chains with rewards.