Stopped Random Walks

Classical probability theory provides information about random walks after a fixed number of steps. For applications, however, it is more natural to consider random walks evaluated after a random number of steps. Stopped Random Walks: Limit Theorems and Applications shows how this theory can be used to prove limit theorems for renewal counting processes, first passage time processes, and certain two-dimensional random walks, as well as how these results may be used in a variety of applications.The present second edition offers updated content and an outlook on further results, extensions and generalizations. A new chapter introduces nonlinear renewal processes and the theory of perturbed random walks, which are modeled as random walks plus "noise".This self-contained research monograph is motivated by numerous examples and problems. With its concise blend of material and over 300 bibliographic references, the book provides a unified and fairly complete treatment of the area. The book may be used in the classroom as part of a course on "probability theory", "random walks" or "random walks and renewal processes", as well as for self-study.From the reviews:"The book provides a nice synthesis of a lot of useful material."American Mathematical Society".[a] clearly written book, useful for researcher and student."Zentralblatt MATH

This new edition offers updated content and an outlook on further results, extensions and generalizations on stopped random walks. It also contains a new chapter on perturbed random walks and nearly 100 new bibliographic references.

Classical probability theory provides information about random walks after a fixed number of steps. For applications, however, it is more natural to consider random walks evaluated after a random number of steps. Examples are sequential analysis, queuing theory, storage and inventory theory, insurance risk theory, reliability theory, and the theory of contours. Stopped Random Walks: Limit Theorems and Applications shows how this theory can be used to prove limit theorems for renewal counting processes, first passage time processes, and certain two-dimenstional random walks, and to how these results are useful in various applications.This second edition offers updated content and an outlook on further results, extensions and generalizations. A new chapter examines nonlinear renewal processes in order to present the analagous theory for perturbed random walks, modeled as a random walk plus "noise."

Preface.- Notations and Symbols.- Introduction.- Limit Theorems for Stopped Random Walks.- Renewal Processes and Random Walks.- Renewal Theory for Random Walks with Positive Drift.- Generalizations and Extensions.- Functional Limit Theorems.- Perturbed Random Walks.- Appendix A: Some Facts from Probability Theory.- Appendix B: Some Facts about Regularly Varying Functions.- Bibliography.- Index.