Gut A. Stopped Random Walks: Limit Theorems and Applications

Second Edition. — Springer, 2009. — 263 p. — ISBN: 978-0387878348, e-ISBN: 978-0387878355.
Series: 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, queuing theory, storage and inventory theory, insurance risk theory, reliability theory, and the theory of contours. 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-dimenstional 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."Limit Theorems for Stopped Random Walks.
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.
Renewal Processes and Random Walks.
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.
Renewal Theory for Random Walks with Positive Drift.
Ladder Variables, Finiteness of Moments, The Strong Law of Large Numbers,
The Central Limit Theorem, Renewal Theorems, Uniform Integrability, Moment Convergence,
Further Results on Eν(t) and Varν(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.
Generalizations and Extensions.
A Stopped Two-Dimensional Random Walk, Some Applications, Chromatographic Methods,
Motion of Water in a River, The Alternating Renewal Process, Cryptomachines,
Age Replacement Policies, Age Replacement Policies; Cost Considerations,
Random Replacement Policies, Counter Models, Insurance Risk Theory,
The Queueing System M/G/1, The Waiting Time in a Roulette Game, A Curious (?) Problem,
The Maximum of a Random Walk with Positive Drift, First Passage Times Across General Boundaries.
Functional Limit Theorems.
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,
Perturbed Random Walks.
Limit Theorems; the General Case, Limit Theorems: the Case Z_n = n g(Y_n),
Convergence Rates, Finiteness of Moments: the General Case,
Finiteness of Moments: the Case Z_n = n g(Y_n),
Moment Convergence: the General Case, Moment Convergence; the Case Z_n = n g(Y_n), Examples,
Stopped Two-Dimensional Perturbed Random Walks, Case Z_n = n g(Y_n),
An Application, Remarks on Further Results and Extensions.
A. Some Facts from Probability Theory:
Convergence of Moments Uniform Integrability, Moment Inequalities for Martingales,
Convergence of Probability Measures, Strong Invariance Principles, Problems.
B. Some Facts about Regularly Varying Functions:
Introduction and Definitions, Some Results.