by Allan Gut

I. Limit Theorems for Stopped Random Walks

- Introduction
- a.s. Convergence and Convergence in Probability
- Anscombe's Theorem
- Moment Convergence in the Strong Law and the Central Limit Theorem
- Moment Inequalities
- Uniform Integrability
- Moment Convergence
- The Stopping Summand
- The Law of the Iterated Logarithm
- Complete Convergence and Convergence Rates
- Problems

II. Renewal Processes and Random Walks

- Introduction
- Renewal Processes; Introductory Examples
- Renewal Processes; Definition and General Facts
- Renewal Theorems
- Limit Theorems
- The Residual Lifetime
- Further Results
- Random Walks; Introduction and Classifications
- Ladder Variables
- The Maximum and the Minimum of a Random Walk
- Representation Formulas for the Maximum
- Limit Theorems for the Maximum

III. Renewal Theory for Random Walks with Positive Drift

- Introduction
- Ladder Variables
- Finiteness of Moments
- The Strong Law of Large Numbers
- The Central Limit Theorem
- Renewal Theorems
- Uniform Integrability
- Moment Convergence
- Further Results on
*E v(t)*and*Var v(t)* - The Overshoot
- The Law of the Iterated Logarithm
- Complete Convergence and Convergence Rates
- Applications to the Simple Random Walk
- Extensions to the Non-I.I.D. Case
- Problems

IV. Generalizations and Extensions

- Introduction
- A Stopped Two-Dimensional Random Walk
- Some Applications
- The Maximum of a Random Walk with Positive Drift
- First Passage Times Across General Boundaries

V. Functional Limit Theorems

- Introduction
- An Anscombe-Donsker Invariance Principle
- First Passage Times for Random Walks with Positive Drift
- A Stopped Two-Dimensional Random Walk
- The Maximum of a Random Walk with Positive Drift
- First Passage Times Across General Boundaries
- The Law of the Iterated Logarithm
- Further Results

Appendix A. Some Facts From Probability Theory

- Convergence of Moments. Uniform Integrability.
- Moment Inequalities for Martingales
- Convergence of Probability Measures
- Strong Invariance Principles
- Problems

Appendix B. Some Facts about Regularly Varying Functions

- Introduction and Definitions
- 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.