Spitzer F. Principles of Random Walk

Second Edition. — Springer, 1976. Printing 2001. — 422 p. — ISBN: 0387951547."This book certainly covers almost all major topics in the theory of random walk. It will be invaluable to both pure and applied probabilists, as well as to many people in analysis. References for the methods and results involved are very good. A useful interdependence guide is given. Excellent choice is made of examples, which are mostly concerned with very concrete calculations. Each chapter contains complementary material in the form of remarks, examples and problems which are often themselves interesting theorems." (T. Watanabe, Mathematical Reviews)The Classification of Random Walk.Periodicity and recurrence behavior.
Some measure theory.
The range of a random walk.
The strong ratio theorem.
Harmonic Analysis.
Characteristic functions and moments.
Periodicity.
Recurrence criteria and examples.
The renewal theorem.
Two-Dimensional Recurrent Random Walk.
Generalities.
The hitting probabilities of a finite set.
The potential kernel A(x,y).
Some potential theory.
The Green function of a finite set.
Simple random walk in the plane.
The time dependent behavior.
Random Walk on a Half-Line.
The hitting probability of the right half-line.
Random walk with finite mean.
The Green function and the gambler's ruin problem.
Fluctuations and the arc-sine law.
Random Walk on a Interval.
Simple random walk.
The absorption problem with mean zero, finite variance.
The Green function for the absorption problem.
Transient Random Walk.
The Green function G(x,y).
Hitting probabilities.
Random walk in three-space with mean zero and finite second moments.
Applications to analysis.
Recurent Random Walk.
The existence of the one-dimensional potential kernel.
The asymptotic behavior of the potential kernel.
Hitting probabilities and the Green function.
The uniqueness of the recurrent potential kernel.
The hitting time of a single point.