Optimal bounds for bitsizes of stationary distributions in finite Markov chains
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.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.04976
 Bibcode:
 2021arXiv210904976S
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Computer Science and Game Theory;
 Mathematics  Probability
 EPrint:
 15 pages, 3 figures