Principles of Random Walk

1. Front Matter

   Pages i-xiii

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2. The Classification of Random Walk

   • Frank Spitzer

   Pages 1-53

3. Harmonic Analysis

   • Frank Spitzer

   Pages 54-104

4. Two-Dimensional Recurrent Random Walk

   • Frank Spitzer

   Pages 105-173

5. Random Walk on a Half-Line

   • Frank Spitzer

   Pages 174-236

6. Random Walk on an Interval

   • Frank Spitzer

   Pages 237-273

7. Transient Random Walk

   • Frank Spitzer

   Pages 274-342

8. Recurrent Random Walk

   • Frank Spitzer

   Pages 343-394

9. Back Matter

   Pages 395-408

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In this edition a large number of errors have been corrected, an occasional proof has been streamlined, and a number of references are made to recent pro­ gress. These references are to a supplementary bibliography, whose items are referred to as [S1] through [S26]. A thorough revision was not attempted. The development of the subject in the last decade would have required a treatment in a much more general con­ text. It is true that a number of interesting questions remain open in the concrete setting of random walk on the integers. (See [S 19] for a recent survey). On the other hand, much of the material of this book (foundations, fluctuation theory, renewal theorems) is now available in standard texts, e.g. Feller [S9], Breiman [S1], Chung [S4] in the more general setting of random walk on the real line. But the major new development since the first edition occurred in 1969, when D. Ornstein [S22] and C. J. Stone [S26] succeeded in extending the recurrent potential theory in· Chapters II and VII from the integers to the reals. By now there is an extensive and nearly complete potential theory of recurrent random walk on locally compact groups, Abelian ( [S20], [S25]) as well as non­ Abelian ( [S17], [S2] ). Finally, for the non-specialist there exists now an unsurpassed brief introduction to probabilistic potential theory, in the context of simple random walk and Brownian motion, by Dynkin and Yushkevich [S8].

• Markov chain
• probability
• probability theory
• random walk

From the reviews:

"...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)

From the reviews of the second edition:

"The most valuable new feature of the second edition is a supplementary bibliography covering results obtained from 1964 to 1976, which have been carefully included into the text. The publication of the second printing now encourages the reader to reconstruct the trains of thought of the founders of the theory of random walk. … For those knowing already a little bit about the theory this book is an invaluable source of ideas, impressive connections and results." (Markus Reiss, Zentralblatt MATH, Vol. 979, 2002)

• Department of Mathematics, Cornell University, Ithaca, USA

  Frank Spitzer

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