Stopped Random Walks: Limit Theorems and Applications
by Allan Gut

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

From the cover of the book:

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 contours. Stopped Random Walks: Limit Theorems and Applications, the first unified treatment of this subject, 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.