Stopped Random Walks: Limit Theorems and Applications (Applied Probability) (v. 5) (Hardcover)

Product Description
Classical probability theory provides information about random walks after a fixed number of steps. For applications it is more natural to consider random walks evaluated after random number of steps. This book offers a unified treatment of the subject and 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 how these results are useful in various applications.

About the Author
Dr. Allan Gut is a professor of mathematical statistics at Uppsala University in Sweden. He has published many numerous articles, and has authored and co-authored six books, four of which were published by Springer. Three of those books, including the first edition of this book, have sold out, and Probability: A Graduate Course, published in 2005, is selling well. --This text refers to the Hardcover edition.
  

I. Limit Theorems for Stopped Random Walks

1. Introduction
2. a.s. Convergence and Convergence in Probability
3. Anscombe's Theorem
4. Moment Convergence in the Strong Law and the Central Limit Theorem
5. Moment Inequalities
6. Uniform Integrability
7. Moment Convergence
8. The Stopping Summand
9. The Law of the Iterated Logarithm
10. Complete Convergence and Convergence Rates
11. Problems

II. Renewal Processes and Random Walks

1. Introduction
2. Renewal Processes; Introductory Examples
3. Renewal Processes; Definition and General Facts
4. Renewal Theorems
5. Limit Theorems
6. The Residual Lifetime
7. Further Results
8. Random Walks; Introduction and Classifications
9. Ladder Variables
10. The Maximum and the Minimum of a Random Walk
11. Representation Formulas for the Maximum
12. Limit Theorems for the Maximum

III. Renewal Theory for Random Walks with Positive Drift

1. Introduction
2. Ladder Variables
3. Finiteness of Moments
4. The Strong Law of Large Numbers
5. The Central Limit Theorem
6. Renewal Theorems
7. Uniform Integrability
8. Moment Convergence
9. Further Results on E v(t) and Var v(t)
      
10. The Overshoot
11. The Law of the Iterated Logarithm
12. Complete Convergence and Convergence Rates
13. Applications to the Simple Random Walk
14. Extensions to the Non-I.I.D. Case
15. Problems

IV. Generalizations and Extensions

1. Introduction
2. A Stopped Two-Dimensional Random Walk
3. Some Applications
4. The Maximum of a Random Walk with Positive Drift
5. First Passage Times Across General Boundaries

V. Functional Limit Theorems

1. Introduction
2. An Anscombe-Donsker Invariance Principle
3. First Passage Times for Random Walks with Positive Drift
4. A Stopped Two-Dimensional Random Walk
5. The Maximum of a Random Walk with Positive Drift
6. First Passage Times Across General Boundaries
7. The Law of the Iterated Logarithm
8. Further Results

Appendix A. Some Facts From Probability Theory

1. Convergence of Moments. Uniform Integrability.
2. Moment Inequalities for Martingales
3. Convergence of Probability Measures
4. Strong Invariance Principles
5. Problems

Appendix B. Some Facts about Regularly Varying Functions

1. Introduction and Definitions
2. Some Results