Coupling for Renewal Schemes

Dmitrii Silvestrov
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Dmitrii Silvestrov: Stockholm University, Department of Mathematics

Chapter Chapter 8 in Coupling and Ergodic Theorems for Semi-Markov-Type Processes I, 2025, pp 315-373 from Springer

Abstract: Abstract In Chap. 8 , the coupling algorithm of exact coupling for renewal schemes is described. It is based on the “gluing” of blocks in the alternating sequence of approaching and synchronizing blocks of renewal points up to the first successful synchronization of renewal points and the following continuation of both renewal schemes using one sequence of renewal points. We also give explicit upper bounds for coupling tail probabilities and power and exponential moments of coupling times. We show that the coupling time for renewal schemes can be represented as the first hitting time to some domain for a semi-Markov process associated with the coupled renewal schemes and give various lower bounds for coupling probability and upper bounds for power and exponential moments of coupling times for renewal schemes using the method of test functions.

Date: 2025
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DOI: 10.1007/978-3-031-89311-7_8

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