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Stopped Random Walks

Limit Theorems and Applications

By: Allan Gut

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Hardcover | 27 February 2009 | Edition Number 2

At a Glance

Hardcover
280 Pages

Edition Type
Revised

Dimensions(cm)
24.41 x 16.99 x 1.75

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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

Industry Reviews

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)

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Non-Fiction Mathematics Probability & Statistics Optimisation Linear Programming Business & Management Operational Research Management & Management Techniques

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Paperback

Published: 15th December 2010

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Preface to the 1st edition	p. v
Preface to the 2nd edition	p. vii
Notation and Symbols	p. xiii
Introduction	p. 1
Limit Theorems for Stopped Random Walks	p. 9
Introduction	p. 9
a.s. Convergence and Convergence in Probability	p. 12
Anscombe's Theorem	p. 16
Moment Convergence in the Strong Law and the Central Limit Theorem	p. 18
Moment Inequalities	p. 21
Uniform Integrability	p. 30
Moment Convergence	p. 39
The Stopping Summand	p. 42
The Law of the Iterated Logarithm	p. 44
Complete Convergence and Convergence Rates	p. 45
Problems	p. 47
Renewal Processes and Random Walks	p. 49
Introduction	p. 49
Renewal Processes; Introductory Examples	p. 50
Renewal Processes; Definition and General Facts	p. 51
Renewal Theorems	p. 54
Limit Theorems	p. 57
The Residual Lifetime	p. 61
Further Results	p. 64
	p. 64
	p. 65
	p. 65
	p. 66
	p. 66
	p. 66
Random Walks; Introduction and Classifications	p. 66
Ladder Variables	p. 69
The Maximum and the Minimum of a Random Walk	p. 71
Representation Formulas for the Maximum	p. 72
Limit Theorems for the Maximum	p. 74
Renewal Theory for Random Walks with Positive Drift	p. 79
Introduction	p. 79
Ladder Variables	p. 82
Finiteness of Moments	p. 83
The Strong Law of Large Numbers	p. 88
The Central Limit Theorem	p. 91
Renewal Theorems	p. 93
Uniform Integrability	p. 96
Moment Convergence	p. 98
Further Results on E¿(t) and Var ¿(t)	p. 100
The Overshoot	p. 103
The Law of the Iterated Logarithm	p. 108
Complete Convergence and Convergence Rates	p. 109
Applications to the Simple Random Walk	p. 109
Extensions to the Non-I.I.D. Case	p. 112
Problems	p. 112
Generalizations and Extensions	p. 115
Introduction	p. 115
A Stopped Two-Dimensional Random Walk	p. 116
Some Applications	p. 126
Chromatographic Methods	p. 126
Motion of Water in a River	p. 129
The Alternating Renewal Process	p. 129
Cryptomachines	p. 130
Age Replacement Policies	p. 130
Age Replacement Policies; Cost Considerations	p. 132
Random Replacement Policies	p. 132
Counter Models	p. 132
Insurance Risk Theory	p. 133
The Queueing System M/G/1	p. 134
The Waiting Time in a Roulette Game	p. 134
A Curious (?) Problem	p. 136
The Maximum of a Random Walk with Positive Drift	p. 136
First Passage Times Across General Boundaries	p. 141
Functional Limit Theorems	p. 157
Introduction	p. 157
An Anscombe–Donsker Invariance Principle	p. 157
First Passage Times for Random Walks with Positive Drift	p. 162
A Stopped Two-Dimensional Random Walk	p. 167
The Maximum of a Random Walk with Positive Drift	p. 169
First Passage Times Across General Boundaries	p. 170
The Law of the Iterated Logarithm	p. 172
Further Results	p. 174
Perturbed Random Walks	p. 175
Introduction	p. 175
Limit Theorems; the General Case	p. 178
Limit Theorems; the Case <$>Z_n = n cdot g(bar Y_n)<$>	p. 183
Convergence Rates	p. 190
Finiteness of Moments; the General Case	p. 190
Finiteness of Moments; the Case <$>Z_n = n cdot g (bar Y_n)<$>	p. 194
Moment Convergence; the General Case	p. 198
Moment Convergence; the Case <$>Z_n = n cdot g(bar Y_n)<$>	p. 200
Examples	p. 202
Stopped Two-Dimensional Perturbed Random Walks	p. 205
The Case <$>Z_n = n cdot g(bar Y_n)<$>	p. 209
An Application	p. 211
Remarks on Further Results and Extensions	p. 214
Problems	p. 221
Some Facts from Probability Theory	p. 223
Convergence of Moments. Uniform Integrability	p. 223
Moment Inequalities for Martingales	p. 225
Convergence of Probability Measures	p. 229
Strong Invariance Principles	p. 234
Problems	p. 235
Some Facts about Regularly Varying Functions	p. 237
Introduction and Definitions	p. 237
Some Results	p. 238
References	p. 241
Index	p. 257
Table of Contents provided by Ingram. All Rights Reserved.

Stopped Random Walks

Limit Theorems and Applications

At a Glance

Hardcover

Industry Reviews

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