Overview

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 withand 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.

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9783319840857

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The book also has an extensive bibliography with useful references to the relevant chapters. The book is well documented as one can see also looking at the bibliographic comments and the bibliography. The material covered in the book is broad in its scope, the exposition is lucid and friendly. Thus, the text will be of considerable interest for university professors and students. (Zdzislaw Rychlik, zbMATH 1382.60004, 2018)

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