zbMATH — the first resource for mathematics

Strong stable Markov chains. (English) Zbl 0874.60082
Utrecht: VSP. Kiev: TBiMC, 138 p. (1996).
The monograph contains a new approach for the study of discrete time general Markov chains evolved by the author which is based on the perturbation theory of linear operators in Banach spaces. In particular, special norms for transition operators are constructed that allow the author to separate their maximal eigenvalues from 1. The lack of such a property for general Markov chains is a usual obstacle in their investigation. The following general topics are considered: (i) Uniform ergodicity: criteria, convergence rate, asymptotic expansions; (ii) Continuity (stability, in the author’s terms) of Markov chains with respect to perturbations of their transition operators: criteria, asymptotic expansions, continuity bounds. Furthermore, the following particular problems are treated: (i) Uniform consolidation theorem; (ii) Estimates of exponential asymptotics of “rare” stopping times; (iii) Continuity estimates in queueing; (iv) Inequalities in the Rényi theorem. Although sometimes the material looks like a collection of papers, the book contains many new and prominent results which successfully can be used in both theoretical and applied fields.

60K25 Queueing theory (aspects of probability theory)
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)