Synchronizing of Shifted Renewal Schemes

Dmitrii Silvestrov
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Dmitrii Silvestrov: Stockholm University, Department of Mathematics

Chapter Chapter 7 in Coupling and Ergodic Theorems for Semi-Markov-Type Processes I, 2025, pp 279-313 from Springer

Abstract: Abstract In Chap. 10.1007/978-3-031-89311-7_7 , we present the synchronizing (coupling) algorithm for shifted renewal points of two renewal schemes and give explicit lower bounds for the corresponding coupling probabilities and upper bounds for power and exponential moments for deviations of non-coupled shifted renewal points. These results are based on relations describing maximal coupling for a nonnegative random variable with a prescribed distribution function and its shifted version. The explicit expressions for maximal coupling probability and two-dimensional distribution with maximal coupling probability are given. Also, a stochastic representation for a random vector with maximal coupling probability and a formula connecting the maximal coupling probability with the variational distance for the corresponding marginal distribution functions are given. Continuity and other features of the maximal coupling probability (as a function of the shift parameter) are described.

Date: 2025
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DOI: 10.1007/978-3-031-89311-7_7

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