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Coupling and Ergodic Theorems for Regenerative Processes

In: Coupling and Ergodic Theorems for Semi-Markov-Type Processes I

Author

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  • Dmitrii Silvestrov

    (Stockholm University, Department of Mathematics)

Abstract

In Chap. 9, we describe the coupling algorithm for regenerative processes and present ergodic theorems for regenerative processes with explicit upper bounds for convergence rates in these theorems. We present alternative equivalent definitions of a regenerative process with a transition period, consider renewal equations for one-dimensional distributions of the regenerative process, show that the characteristics of the regenerative process related to its transition period can be chosen in the way providing stationarity of one-dimensional distributions of the regenerative process, and present a coupling algorithm for regenerative processes. These results let us obtain the main ergodic for regenerative processes with explicit power and exponential upper bounds for variational distances between one-dimensional distributions of regenerative processes and the corresponding stationary distributions, using upper bounds for power and exponential moments of coupling times for renewal schemes.

Suggested Citation

  • Dmitrii Silvestrov, 2025. "Coupling and Ergodic Theorems for Regenerative Processes," Springer Books, in: Coupling and Ergodic Theorems for Semi-Markov-Type Processes I, chapter 0, pages 375-414, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-89311-7_9
    DOI: 10.1007/978-3-031-89311-7_9
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