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The paper introduces and explores the notions of ergodicity index and ergodicity exponent of a set of stochastic matrices. For the ergodicity exponent a sharp upper bound is obtained. A particular case of this bound is the well-known Paz bound. Also a connection between the ergodicity index and the Protasov–Voynov imprimitivity index is established. [ABSTRACT FROM AUTHOR]

The paper proves that an indecomposable nonnegative matrix is chainable if and only if it satisfies Kolmogorov's condition, under which the Markov chain determined by a stochastic matrix obeys the multidimensional local limit theorem. [ABSTRACT FROM AUTHOR]

A description of nonnegative chainable matrices, based on fully indecomposable matrices, is given. The notion of the chainable rank of a nonnegative matrix is introduced and investigated. Properties of chainable matrices are used in studying stochastic Sarymsakov matrices. [ABSTRACT FROM AUTHOR]

The paper studies the notion of imprimitivity index of a semigroup of nonnegative matrices, introduced by Protasov and Voynov. A new characterization of the imprimitivity index in terms of the scrambling rank of a nonnegative matrix is suggested. Based on this characterization, an independent combinatorial proof of the Protasov-Voynov theorem on the interrelation between the imprimitivity index of a semigroup of stochastic matrices and the spectral properties of matrices in the semigroup is presented. [ABSTRACT FROM AUTHOR]

The paper suggests a new linear-algebraic proof of a formula for the Perron vector (stationary distribution) of a stochastic matrix, known in the theory of Markov chains. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]