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.

中文翻译：

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有限马尔可夫链中平稳分布的位大小的最佳边界

具有有理项的不可约随机矩阵具有由有理数向量给出的平稳分布。我们给出了这个向量条目的最小公分母的上限。这种边界用于研究解决随机平均收益博弈的算法的复杂性。它们通常是使用 Hadamard 不等式推导出来的，但这会导致次优结果。我们用马尔可夫链树公式代替 Hadamard 不等式以获得最优边界。我们还调整了我们的方法以获得有限马尔可夫链的吸收概率以及带奖励的马尔可夫链的增益和偏置向量的界限。