Perturbed Semi-Markov Type Processes II: Ergodic Theorems for Multi-Alternating Regenerative Processes

✍ Scribed by Dmitrii Silvestrov

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 420
• Edition: 1st ed. 2022
• Category: Library

✦ Synopsis

This book is the second volume of a two-volume monograph devoted to the study of limit and ergodic theorems for regularly and singularly perturbed Markov chains, semi-Markov processes, and multi-alternating regenerative processes with semi-Markov modulation.
The second volume presents a complete classification of ergodic theorems for alternating regenerative processes, including more than twenty-five such theorems. The text addresses new asymptotic recurrent algorithms of phase space reduction for multi-alternating regenerative processes modulating by regularly and singularly perturbed finite semi-Markov processes. It also features a new study of super-long, long, and short time ergodic theorems for these processes.
The book also contains a comprehensive bibliography of major works in the field. It provides an effective reference for both graduate students as well as theoretical and applied researchers studying stochastic processes and their applications.

✦ Table of Contents

Preface
Contents
List of Symbols
1 Introduction
1.1 Part I: Ergodic Theorems for Perturbed Alternating Regenerative Processes
1.1.1 Part I: Contents, Examples, Models, and Results
1.1.2 Part I: Contents by Chapters
1.2 Part II: Ergodic Theorems for Perturbed Multi-Alternating Regenerative Processes
1.2.1 Part II: Contents, Examples, Models, and Results
1.2.2 Part II: Contents by Chapters
1.3 Appendices and Conclusion
1.3.1 Appendix A: Perturbed Renewal Equation
1.3.2 Appendix B: Supplementary Asymptotic Results
1.3.3 Appendix C: Methodological and Bibliographical Notes
1.3.4 Conclusion
Part I Ergodic Theorems for Perturbed Alternating Regenerative Processes
2 Ergodic Theorems for Perturbed Regenerative Processes
2.1 Regenerative Processes with Regenerative Lifetimes
2.1.1 Regenerative Processes with Regenerative Lifetimes
2.1.2 Perturbation Conditions for Regenerative Processes with Regenerative Lifetimes
2.2 Ergodic Theorems for Perturbed Regenerative Processes with Regenerative Lifetimes
2.2.1 Ergodic Theorems for Perturbed Regenerative Processes
2.2.2 Ergodic Theorems for Perturbed Regenerative Processes with Modified Regenerative Lifetimes
3 Perturbed Alternating Regenerative Processes
3.1 Alternating Regenerative Processes
3.1.1 Alternating Regenerative Processes
3.1.2 Perturbation Conditions for Alternating Regenerative Processes
3.2 Regularly, Singularly, and Super-Singularly Perturbed Alternating Regenerative Processes
3.2.1 Regular, Singular, and Super-Singular Perturbation Models for Alternating Regenerative Processes
3.2.2 Super-Long, Long, and Short Time Ergodic Theorems for Perturbed Alternating Regenerative Processes
3.3 Time Compression and Aggregation of Regeneration Times for Perturbed Alternating Regenerative Processes
3.3.1 Time Compression for Perturbed Regenerative and Alternating Regenerative Processes
3.3.2 Aggregation of Regeneration Times and Embedded Regenerative Processes
3.3.3 Embedded Regenerative Processes and Ergodic Theorems for Perturbed Alternating Regenerative Processes
4 Ergodic Theorems for Regularly Perturbed Alternating Regenerative Processes
4.1 Regularly Perturbed Alternating Regenerative Processes and Embedded Regenerative Processes of the First Type
4.1.1 Regularly and Semi-regularly Perturbed Alternating Regenerative Processes
4.1.2 Embedded Regenerative Processes of the First Type
4.2 Ergodic Theorems for Perturbed Standard Alternating Regenerative Processes
4.3 Ergodic Theorems for Regularly Perturbed Alternating Regenerative Processes
4.4 Ergodic Theorems for Semi-regularly Perturbed Alternating Regenerative Processes
5 Ergodic Theorems for Regularly Perturbed Alternating Regenerative Processes Compressed in Time
5.1 Regularly Perturbed Alternating Regenerative Processes with Degenerating Regeneration Times
5.2 Compression in Time for Regularly Perturbed Alternating Regenerative Processes
6 Super-Long and Long Time Ergodic Theorems for SingularlyPerturbed Alternating Regenerative Processes
6.1 Singularly Perturbed Alternating Regenerative Processes and Aggregation of Regeneration Times
6.1.1 Singularly Perturbed Alternating Regenerative Processes
6.1.2 Embedded Regenerative Processes of the Second Type
6.2 Super-Long Time Ergodic Theorems and Embedded Regenerative Processes
6.3 Long Time Ergodic Theorems for Singularly Perturbed Alternating Regenerative Processes
7 Short Time Ergodic Theorems for Singularly Perturbed AlternatingRegenerative Processes
7.1 Short Time Ergodic Theorems Based on the First Time Compression Factor
7.1.1 Two Types of Time Compression Factors for Perturbed Alternating Regenerative Processes
7.1.2 Ergodic Theorems Based on the First Time Compression Factor
7.2 Short Time Ergodic Theorems Based on the Second Time Compression Factor
7.2.1 First Type Short Time Ergodic Theorems Based on the Second Time Compression Factor
7.2.2 Second Type Short Time Ergodic Theorems Based on the Second Time Compression Factor
7.2.3 Third Type Short Time Ergodic Theorems Based on the Second Time Compression Factor
8 Ergodic Theorems for Singularly Perturbed Alternating Regenerative Processes Compressed in Time
8.1 Singularly Perturbed Alternating Regenerative Processes with Degenerating Regeneration Times
8.1.1 Super-Long and Long Time Ergodic Theorems
8.1.2 Short Time Ergodic Theorems
8.2 Compression in Time for Singularly Perturbed Alternating Regenerative Processes
9 Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes
9.1 Super-Long, Long, and Short Time Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes
9.1.1 Super-Singularly Perturbed Alternating Regenerative Processes
9.1.2 Super-Long Time Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes
9.1.3 Long Time Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes
9.1.4 Short Time Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes
9.2 Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes Compressed in Time
9.2.1 Super-Singularly Perturbed Alternating Regenerative Processes with Degenerating Regeneration Times
9.2.2 Compression in Time for Super-Singularly Perturbed Alternating Regenerative Processes
9.3 Generalisations and Classification of Ergodic Theorems for Perturbed Alternating Regenerative Processes
9.3.1 Generalisations of Ergodic Theorems for Perturbed Alternating Regenerative Processes
9.3.2 Classification of Ergodic Theorems for Perturbed Alternating Regenerative Processes
Part II Ergodic Theorems for Perturbed Multi-Alternating Regenerative Processes
10 Perturbed Multi-Alternating Regenerative Processes
10.1 Multi-Alternating Regenerative Processes
10.1.1 Definition of Multi-Alternating Regenerative Processes
10.1.2 Perturbation Conditions for Multi-Alternating Regenerative Processes
10.2 Multi-Alternating Regenerative Processes with Removed of Virtual Transitions
10.2.1 Procedure of Total Removing of Virtual Transitions for Modulating Semi-Markov Processes
10.2.2 Procedure of Partial Removing of Virtual Transitions for Modulating Semi-Markov Processes
10.3 Multi-Alternating Regenerative Processes with Reduced Modulating Semi-Markov Processes
10.3.1 Procedure of One-State Reduction of Phase Space for Modulating Semi-Markov Processes
10.3.2 Modified Procedure of One-State Reduction of Phase Space for Modulating Semi-Markov Processes
11 Time–Space Aggregation of Regeneration Times for PerturbedMulti-Alternating Regenerative Processes
11.1 Multi-Alternating Regenerative Processes with Removed Virtual Transitions and Reduced Phase Space for Modulating Semi-Markov Processes
11.1.1 Multi-Alternating Regenerative Processes with Totally or Partially Removed Virtual Transitions for Modulating Semi-Markov Processes
11.1.2 Multi-Alternating Regenerative Processes with Reduced Phase Space of Modulating Semi-Markov Processes
11.2 Time–Space Aggregation for Regeneration Times Based on Total Removing of Virtual Transitions
11.2.1 Algorithm of Time–Space Aggregation for Regeneration Times Based on Total Removing of Virtual Transitions
11.2.2 Summary of Algorithm of Time–Space Aggregation for Regeneration Times Based on Total Removing of Virtual Transitions
11.3 Time–Space Aggregation for Regeneration Times Based on Partial Removing of Virtual Transitions
11.3.1 Algorithm of Time–Space Aggregation for Regeneration Times Based on Partial Removing of Virtual Transitions
11.3.2 Summary of Algorithm of Time–Space Aggregation of Regeneration Times Based on Partial Removing of Virtual Transitions
11.4 Comparison of Recurrent Algorithms of Time–Space Aggregation for Regeneration Times
11.4.1 Algorithms of Time–Space Aggregation of Regeneration Times Based on Total or Partial Removing of Virtual Transitions
11.4.2 Asymptotic Communicative Structure of Phase Spaces for Modulating Semi-Markov Processes for Embedded Alternating Regenerative Processes
12 Embedded Processes for Perturbed Multi-Alternating Regenerative Processes
12.1 Embedded Alternating Regenerative Processes
12.1.1 Embedded Alternating Regenerative Processes Based on Total Removing of Virtual Transitions
12.1.2 Embedded Alternating Regenerative Processes Based on Partial Removing of Virtual Transitions
12.2 Embedded Regenerative Processes
12.2.1 Embedded Regenerative Processes Based on Total Removing of Virtual Transitions
12.2.2 Embedded Regenerative Processes Based on Partial Removing of Virtual Transitions
13 Ergodic Theorems for Perturbed Multi-Alternating Regenerative Processes
13.1 Ergodic Theorems for Regularly Perturbed Multi-Alternating Regenerative Processes
13.1.1 Ergodic Theorems Based on Regularly Perturbed Embedded Alternating Regenerative Processes with Totally Removed Virtual Transitions
13.1.2 Ergodic Theorems Based on Regularly Perturbed Embedded Alternating Regenerative Processes with Partially Removed Virtual Transitions
13.2 Ergodic Theorems for Singularly Perturbed Multi-Alternating Regenerative Processes
13.2.1 Super-Long and Long Time Ergodic Theorems Based on Singularly Perturbed Embedded Alternating Regenerative Processes
13.2.2 Short Time Ergodic Theorems Based on Singularly Perturbed Embedded Alternating Regenerative Processes
13.3 Ergodic Theorems for Perturbed Multi-Alternating Regenerative Processes Based on Embedded Regenerative Processes
13.3.1 Ergodic Theorems Based on Perturbed Embedded Regenerative Processes
13.3.2 Relationship Between Ergodic Theorems Based on Perturbed Embedded Alternating Regenerative Processes and Embedded Regenerative Processes
A Perturbed Renewal Equation
A.1 Renewal Equation and Renewal Theorem
A.1.1 Renewal Equation
A.1.2 Non-Arithmetic Distribution Functions
A.1.3 Renewal Theorem
A.2 Perturbed Renewal Equation and Renewal Theorem
A.2.1 Perturbed Renewal Equation
A.2.2 Renewal Theorem for Perturbed Renewal Equation
A.2.3 Renewal Theorem for Perturbed Renewal Equation with Transition Renewal Period
B Supplementary Asymptotic Results
B.1 Limit Theorems for Stochastic Processes
B.1.1 Limit Theorems for Randomly Stopped Stochastic Processes
B.1.2 Slutsky Theorem and Related Results
B.2 Other Useful Asymptotic Results
B.2.1 Asymptotically Comparable Functions
B.2.2 Convergence of Lebesgue Integrals in the Scheme of Series
C Methodological and Bibliographical Notes
C.1 Methodological Notes
C.2 General Bibliographical Remarks
References
Index