Renewal Theory for Perturbed Random Walks and Similar Processes

Authors: Alexander Iksanov



Provides a thorough discussion of the state-of-the art in the area with a special emphasis on the methods employed

Gives results in a final form and poses a number of open questions at the same time

Discusses numerous examples and applications

This book offers a detailed review of perturbed random walks, perpetuities, and random processes with immigration. Being of major importance in modern probability theory, both theoretical and applied, these objects have been used to model various phenomena in the natural sciences as well as in insurance and finance. The book also presents the many significant results and efficient techniques and methods that have been worked out in the last decade.

The first chapter is devoted to perturbed random walks and discusses their asymptotic behavior and various functionals pertaining to them, including supremum and first-passage time. The second chapter examines perpetuities, presenting results on continuity of their distributions and the existence of moments, as well as weak convergence of divergent perpetuities. Focusing on random processes with immigration, the third chapter investigates the existence of moments, describes long-time behavior and discusses limit theorems, both with and without scaling. Chapters four and five address branching random walks and the Bernoulli sieve, respectively, and their connection to the results of the previous chapters.

With many motivating examples, this book appeals to both theoretical and applied probabilists.

Table of contents

Front Matter
Pages i-xiv

Perturbed Random Walks
Pages 1-41

Perpetuities
Pages 43-86

Random Processes with Immigration
Pages 87-178

Application to Branching Random Walk
Pages 179-189

Application to the Bernoulli Sieve
Pages 191-208

Appendix
Pages 209-236

Back Matter
Pages 237-250

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