Gut Stopped Random Walks

Limit Theorems and Applications
2. Auflage 2009
ISBN: 978-0-387-87835-5
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 267 Seiten

Reihe: Springer Series in Operations Research and Financial Engineering

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, queueing theory, storage and inventory theory, insurance risk theory, reliability theory, and the theory of counters. 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, 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”.

Autoren/Hrsg.

Gut, Allan

Inhaltsverzeichnis

1;Preface to the 1st edition;5
2;Preface to the 2nd edition;6
3;Contents;7
4;Notation and Symbols;10
5;Introduction;11
6;1 Limit Theorems for Stopped Random Walks;19
6.1;1.1 Introduction;19
6.2;1.2 a.s. Convergence and Convergence in Probability;22
6.3;1.3 Anscombe’s Theorem;26
6.4;1.4 Moment Convergence in the Strong Law and the Central Limit Theorem;28
6.5;1.5 Moment Inequalities;31
6.6;1.6 Uniform Integrability;40
6.7;1.7 Moment Convergence;49
6.8;1.8 The Stopping Summand;52
6.9;1.9 The Law of the Iterated Logarithm;54
6.10;1.10 Complete Convergence and Convergence Rates;55
6.11;1.11 Problems;57
7;2 Renewal Processes and Random Walks;58
7.1;2.1 Introduction;58
7.2;2.2 Renewal Processes; Introductory Examples;59
7.3;2.3 Renewal Processes; Definition and General Facts;60
7.4;2.4 Renewal Theorems;63
7.5;2.5 Limit Theorems;66
7.6;2.6 The Residual Lifetime;70
7.7;2.7 Further Results;73
7.8;2.8 Random Walks; Introduction and Classifications;75
7.9;2.9 Ladder Variables;78
7.10;2.10 The Maximum and the Minimum of a Random Walk;80
7.11;2.11 Representation Formulas for the Maximum;81
7.12;2.12 Limit Theorems for the Maximum;83
8;3 Renewal Theory for Random Walks with Positive Drift;87
8.1;3.1 Introduction;87
8.2;3.2 Ladder Variables;90
8.3;3.3 Finiteness of Moments;91
8.4;3.4 The Strong Law of Large Numbers;96
8.5;3.5 The Central Limit Theorem;99
8.6;3.6 Renewal Theorems;101
8.7;3.7 Uniform Integrability;104
8.8;3.8 Moment Convergence;106
8.9;3.9 Further Results on E.(t) and Var.(t);108
8.10;3.10 The Overshoot;111
8.11;3.11 The Law of the Iterated Logarithm;116
8.12;3.12 Complete Convergence and Convergence Rates;117
8.13;3.13 Applications to the Simple Random Walk;117
8.14;3.14 Extensions to the Non-I.I.D. Case;120
8.15;3.15 Problems;120
9;4 Generalizations and Extensions;122
9.1;4.1 Introduction;122
9.2;4.2 A Stopped Two-Dimensional Random Walk;123
9.3;4.3 Some Applications;133
9.4;4.4 The Maximum of a Random Walk with Positive Drift;143
9.5;4.5 First Passage Times Across General Boundaries;148
10;5 Functional Limit Theorems;164
10.1;5.1 Introduction;164
10.2;5.2 An Anscombe–Donsker Invariance Principle;164
10.3;5.3 First Passage Times for Random Walks with Positive Drift;169
10.4;5.4 A Stopped Two-Dimensional Random Walk;174
10.5;5.5 The Maximum of a Random Walk with Positive Drift;176
10.6;5.6 First Passage Times Across General Boundaries;177
10.7;5.7 The Law of the Iterated Logarithm;179
10.8;5.8 Further Results;181
11;6 Perturbed Random Walks;182
11.1;6.1 Introduction;182
11.2;6.2 Limit Theorems; the General Case;185
11.3;6.3 Limit Theorems; the Case Zn = n · g( ¯;190
11.4;6.4 Convergence Rates;197
11.5;6.5 Finiteness of Moments; the General Case;197
11.6;6.6 Finiteness of Moments; the Case Zn = n · g(Yn) ¯;201
11.7;6.7 Moment Convergence; the General Case;205
11.8;6.8 Moment Convergence; the Case Zn = n · g( ¯;207
11.9;6.9 Examples;209
11.10;6.10 Stopped Two-Dimensional Perturbed Random Walks;212
11.11;6.11 The Case Zn = n · g( ¯;216
11.12;6.12 An Application;218
11.13;6.13 Remarks on Further Results and Extensions;221
11.14;6.14 Problems;228
12;A Some Facts from Probability Theory;229
12.1;A.1 Convergence of Moments. Uniform Integrability;229
12.2;A.2 Moment Inequalities for Martingales;231
12.3;A.3 Convergence of Probability Measures;235
12.4;A.4 Strong Invariance Principles;240
12.5;A.5 Problems;241
13;B Some Facts about Regularly Varying Functions;243
13.1;B.1 Introduction and Definitions;243
13.2;B.2 Some Results;244
14;References;246
15;Index;261