Stopped Random Walks: Limit Theorems and Applications (Springer Series in Operations Research and Financial Engineering)

Allan Gut

Hardcover

ISBN 10: 0387878343 ISBN 13: 9780387878348

Publisher: Springer, 2009

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."

Review:

From the reviews of the second edition:

"Stopped random walks occur in sequential analysis renewal theory and queueing theory several applications are discussed in the text. ... The book under review is the second edition of a book first published in 1988 ... . lengthy bibliography from the first edition has been brought up to date. ... an excellent reference and its material is still worthy of study." (Thomas Polaski, Mathematical Reviews, Issue 2010 f)

"This is definitely a book for the specialist in the field. ... It would suit an academic or a researcher ... seeking to use random walks as a tool for some real-world problem. ... the material is very thorough and there are plentiful references. All results are rigorously established, either by a formal proof or by pointing the reader in the right direction. It will enable any researcher to be right up to date with the latest developments in the field." (F. McGonigal, Journal of the Operational Research Society, Vol. 62 (2), 2011)

From the Back Cover:

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