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Coupling and the Renewal Theorem

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. 12 , the variants of the renewal theorem with explicit upper bounds of convergence rates are given for a renewal equation generated by a distribution function F with an absolutely continuous component and the free term of this equation, which is dominated by the tail probabilities of F. These theorems are also specified for the case of the so-called alternating renewal equations. We show how such theorems can be used for obtaining ergodic theorems for real-valued, time-homogeneous, strongly Markov processes. We also present variants of the renewal theorem with explicit upper bounds for convergence rates for an improper renewal equation. A typical example of applications connected with the so-called quasi-stationary ergodic theorems is discussed. The aforementioned theorems are also illustrated by explicit upper bounds for rates of convergence in the classical Cramér–Lundberg approximation for ruin probabilities for risk processes.

Suggested Citation

  • Dmitrii Silvestrov, 2025. "Coupling and the Renewal Theorem," Springer Books, in: Coupling and Ergodic Theorems for Semi-Markov-Type Processes I, chapter 0, pages 511-547, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-89311-7_12
    DOI: 10.1007/978-3-031-89311-7_12
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