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

Table of Contents

1. Frontmatter

2. Chapter 1. Introduction

   Dmitrii Silvestrov

   Abstract

   The chapter is an introduction to the book devoted to studying ergodic theorems with explicit power and exponential upper bounds for convergence rates for semi-Markov processes and multi-alternating regenerative processes with semi-Markov modulation. These ergodic theorems are obtained using coupling ergodic theorems for regenerative processes presented in the first volume, the method of artificial regeneration, and the method of test functions. The introduction informally presents the main problems, methods, and algorithms that comprise the book’s contents. We give simple examples, illustrated by 12 figures, and try to show the logic and ideas underlying the method of test functions and the method of artificial regeneration and their applications for obtaining ergodic theorems with explicit upper bounds for convergence rates for semi-Markov processes and multi-alternating processes with semi-Markov modulation, as well as to explain the meaning of the results presented in the book.

3. Chapter 2. Summary of Ergodic Theorems for Regenerative Processes

   Dmitrii Silvestrov

   Abstract

   In Chap. 2, we present the summary of coupling ergodic theorems with explicit upper bounds for convergence rates for regenerative processes obtained in the first volume. These theorems are a basis for getting ergodic theorems with explicit upper bounds for convergence rates for semi-Markov processes and multi-alternating regenerative processes with semi-Markov modulation. We give the constructive definitions of a regenerative process with a transition period and thinned regenerative processes, formulate and comment on the weighting absolute continuity condition, which plays a key role in ergodic theorems with explicit upper bounds for convergence rates, and present ergodic theorems explicit power and exponential upper bounds for convergence rates for regenerative processes.

4. Chapter 3. Modifications of Hitting Times

   Dmitrii Silvestrov

   In this chapter, we present necessary and sufficient conditions of finiteness and upper bounds for power and exponential moments for some modifications of Markov generalized hitting times, namely extended Markov generalized hitting times. These upper bounds are obtained using the method of test functions. The extended Markov generalized hitting times appear in applications to semi-Markov processes with distributional atoms, semi-Markov processes admitting artificial regeneration, and multi-alternating regenerative processes modulated by such semi-Markov processes, where they play roles of regeneration times. The moments of extended hitting times are key in getting explicit upper bounds for convergence rates in ergodic theorems for such processes. We derive integral equations for moments of extended Markov generalized hitting times and obtain serial representations for these moments as minimal solutions of these equations, present necessary and sufficient conditions of finiteness, and give recursive upper bounds for power moments and upper bounds for exponential moments of extended generalized hitting times.

5. Chapter 4. Birth-Death-Type Processes

   Dmitrii Silvestrov

   Abstract

   In Chap. 4, we present ergodic theorems with explicit upper bounds for convergence rates for semi-Markov birth-death-type processes, which are regenerative processes with regeneration moments that are sequential return moments into any state, in particular, in state 0. Various upper bounds, respectively, for power and exponential moments of hitting times into state 0 are obtained using the method of test functions and recurrent relations connecting moments of hitting times for semi-Markov birth-death-type processes. Finally, ergodic theorems with explicit power and exponential upper bounds for convergence rates are given.

6. Chapter 5. Semi-Markov Processes with Discrete State Spaces and Embedded Regenerative Processes

   Dmitrii Silvestrov

   Abstract

   In Chap. 5, we present ergodic theorems with explicit upper bounds for convergence rates for semi-Markov processes with discrete state spaces. Such processes are regenerative, and sequential return moments in any state are regeneration moments for them. We present necessary and sufficient conditions of finiteness and give upper bounds for power moments of return times supplemented by some solidarity assertions and necessary and sufficient conditions of finiteness and upper bounds exponential moments of return times using the method of test functions. Then, we obtain ergodic theorems with explicit power and exponential upper bounds for convergence rates for semi-Markov processes with discrete state spaces. Also, we present an example of semi-Markov versions of the Hastings-Metropolis MCMC algorithm and comment on a possible interpretation of the above ergodic theorems applied to the corresponding semi-Markov process as distributional semi-Markov versions of this algorithm.

7. Chapter 6. Ergodic Theorems for Queuing Systems

   Dmitrii Silvestrov

   Abstract

   In Chap. 6, we present ergodic theorems with explicit upper bounds for convergence rates for M/G-type queuing systems. We introduce an M/G-type queuing system with service distributions depending on the queue and obtain explicit upper bounds for power and exponential moments for the duration of the busy period using the method of test functions. Then, we present ergodic theorems with explicit power and exponential upper bounds for convergence rates for M/G-type queuing systems with service distributions depending on the queue.

8. Chapter 7. Semi-Markov Processes with General State Spaces with Atoms

   Dmitrii Silvestrov

   Abstract

   In Chap. 7, we present ergodic theorems with explicit upper bounds for rates of convergence for semi-Markov processes with general state spaces and atom (a state x_• for which the one-state subset \(D_{x_\bullet } = \{ x_\bullet \}\) is recurrent). We introduce a regenerative semi-Markov process with an atom and get relations that express the distribution functions of its regeneration time and duration of the transition period via the distribution functions of the first hitting times into the corresponding atomic set, present necessary and sufficient conditions of finiteness and upper bounds given in terms of test functions for power and exponential moments of regeneration times and the duration of the transition period. We also present variants of defining relations for stationary distributions for semi-Markov processes with general state space and atom and its accompanying Markov process and ergodic theorems with explicit power and exponential upper bounds for convergence rates for such semi-Markov processes.

9. Chapter 8. Semi-Markov Processes with General State Spaces and Distributional Atoms

   Dmitrii Silvestrov

   Abstract

   In Chap. 8, we present ergodic theorems with explicit upper bounds for convergence rates for semi-Markov processes with general state spaces and distributional atoms. We introduce regenerative semi-Markov processes with D-distributional atoms with regeneration moments, which are sequential moments of jumps from states in set D, and derive relations that express the distribution functions of its regeneration time and duration of the transition period via the distribution functions of the first hitting times into the set D. Also, necessary and sufficient conditions of finiteness and upper bounds for power and exponential moments of regeneration times, and the duration of the transition period are given in terms of test functions. Finally, we present variants of defining relations for stationary distributions for a semi-Markov process with general state space and distributional atom and its accompanying Markov processes, and ergodic theorems with explicit power and exponential upper bounds for convergence rates for such semi-Markov processes.

10. Chapter 9. Semi-Markov Processes with General State Spaces and One-Step Artificial Regeneration

    Dmitrii Silvestrov

    Abstract

    In Chap. 9, we present ergodic theorems with explicit upper bounds for convergence rates for semi-Markov processes with general state spaces admitting so-called one-step artificial regeneration. This method is based on adding a special indicator component to a semi-Markov process in such a way that the extended process is a regenerative semi-Markov process with a distributional atom. We introduce semi-Markov processes and their accompanying Markov processes admitting one-step artificial regeneration, describe the semi-Markov variant of the method of artificial regeneration, and construct using this method the corresponding extended semi-Markov process and its accompanying Markov process. Upper bounds for power and exponential moments of the regenerative time and the duration of the transition period for semi-Markov processes with distributional atoms are translated to extended semi-Markov processes resulting from the splitting algorithm applied to semi-Markov processes admitting one-step artificial regeneration. Finally, representations for the corresponding stationary distributions of extended semi-Markov processes and initial semi-Markov processes and ergodic theorems with explicit power and exponential upper bounds for convergence rates for semi-Markov processes and their accompanying Markov processes admitting one-step artificial regeneration are given.

11. Chapter 10. Semi-Markov Processes with General State Spaces and Multi-step Artificial Regeneration

    Dmitrii Silvestrov

    Abstract

    In Chap. 10, we present ergodic theorems with explicit upper bounds for convergence rates for semi-Markov processes with general state spaces, which do not admit a one-step artificial regeneration but do admit a multi-step artificial regeneration. We present the multi-step splitting condition and describe the multi-step splitting algorithm for a semi-Markov process. This algorithm lets us construct a regenerative extended semi-Markov process with a distributional atom for the thinned version of a semi-Markov process. Upper bounds for power and exponential moments of the regenerative time and the duration of the transition period for extended semi-Markov processes resulting from the one-step splitting algorithm are translated to extended semi-Markov processes resulting from the multi-step splitting algorithm. Finally, representations for the corresponding stationary distributions of extended semi-Markov processes and initial semi-Markov processes and ergodic theorems with explicit power and exponential upper bounds for convergence rates for semi-Markov processes and their accompanying Markov processes admitting multi-step artificial regeneration are given.

12. Chapter 11. Multi-alternating Regenerative Processes with Semi-Markov Modulation

    Dmitrii Silvestrov

    Abstract

    In Chap. 11, we present ergodic theorems with explicit upper bounds for convergence rates for multi-alternating regenerative processes with semi-Markov modulation. These processes generalize naturally both regenerative and semi-Markov processes. We introduce multi-alternating regenerative processes modulated by semi-Markov processes and give renewal-type equations for distributions of multi-alternating regenerative processes and relations connecting their distributions and the distributions of accompanying Markov processes for modulating 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 semi-Markov processes with atoms and for multi-alternating regenerative processes modulated by semi-Markov processes admitting artificial regeneration.

13. Chapter 12. Multi-alternating Regenerative Processes Modulated by Uniformly Recurrent Semi-Markov Processes

    Dmitrii Silvestrov

    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.

14. Backmatter