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Multi-alternating Regenerative Processes Modulated by Uniformly Recurrent Semi-Markov Processes

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

Author

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  • Dmitrii Silvestrov

    (Stockholm University, Department of Mathematics)

Abstract

In this chapter, we present ergodic theorems with explicit upper bounds for convergence rates for multi-alternating regenerative processes modulated by uniformly recurrent semi-Markov processes. We consider uniformly recurrent semi-Markov processes and get uniform (in the class of all initial distributions) upper bounds for tail probabilities and power and exponential moments for hitting times for such semi-Markov processes. We give two series of ergodic theorems with explicit power and exponential upper bounds for convergence rates for multi-alternating regenerative processes modulated by uniformly recurrent semi-Markov processes with atoms and for multi-alternating regenerative processes modulated by uniformly recurrent semi-Markov processes admitting artificial regeneration. Also, applications of ergodic theorems for multi-alternating regenerative processes with semi-Markov modulation for getting variants of semi-Markov renewal theorem with explicit power and exponential upper bounds for convergence rates are discussed.

Suggested Citation

  • Dmitrii Silvestrov, 2025. "Multi-alternating Regenerative Processes Modulated by Uniformly Recurrent Semi-Markov Processes," Springer Books, in: Coupling and Ergodic Theorems for Semi-Markov-Type Processes II, chapter 0, pages 455-519, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-89315-5_12
    DOI: 10.1007/978-3-031-89315-5_12
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