The first edition of this book appeared in 1950 [MR0038583]. The second edition contains the same material but rearranged and with substantial additions. Especially the third chapter on fluctuations in coin tossing and random walks contains much new and gives a thorough discussion of the arc sine law and related questions. This law is illustrated graphically by the very striking result of a random experiment simulating 10,000 tosses of a coin. Also new is the twelfth chapter on compound distributions and branching processes, which contains a fuller discussion of these topics than the first edition. Among many other changes some should be mentioned explicitly. The theory of recurrent events, which was stressed already in the first edition, is now given still greater dominance. The concept of waiting times is now introduced already in the second chapter. Bivariate generating functions are given a separate section. Processes involving escape (called degenerate processes in the first edition) are approached in a different way making such processes a non-pathological part of the general theory. As in the first edition the exposition is mathematically rigorous and at the same time elegant and lucid. This fascinating book will remain a standard textbook of mathematical probability for many years to come.
Reviewer: Grenander, U. [form MathSciNet]