Probability theory: Independence, interchangeability, martingales.
3rd ed.

*(English)*Zbl 0891.60002
Springer Texts in Statistics. New York, NY: Springer. xxii, 488 p. (1997).

[For a review of the second edition (1988) see Zbl 0652.60001.]

This book gives a complete construction of the probability theory. The measure and integration theories are discussed from the very beginning and all necessary theorems are proved. No prior knowledge of measure theory is assumed. The book deals with e.g. sums of independent random variables, stopping times, optimal stopping, Wald’s equations, interchangeable random variables, elementary renewal equations, central limit theorems, characteristic functions. A great attention is devoted to the martingale theory. Besides the convergence theorems the authors consider the Marcinkiewicz-Zygmund inequality for the sums of independent random variables, its generalization for martingales, i.e. the Burkholder-Davis-Gundy inequality. This last inequality is also proved for convex functions. The section about \(U\)-statistics and some theorems concerning this are added in this third edition.

The presentation of the subject is concise, mathematically rigorous and yet very intelligible and lucid. The considerations are illustrated by many examples. There are also exercises after every section and references after each chapter. The book should be easily adapted and very useful for graduate students. It may serve also as a teaching aid for lecturers at universities and a source of reference for researchers.

This book gives a complete construction of the probability theory. The measure and integration theories are discussed from the very beginning and all necessary theorems are proved. No prior knowledge of measure theory is assumed. The book deals with e.g. sums of independent random variables, stopping times, optimal stopping, Wald’s equations, interchangeable random variables, elementary renewal equations, central limit theorems, characteristic functions. A great attention is devoted to the martingale theory. Besides the convergence theorems the authors consider the Marcinkiewicz-Zygmund inequality for the sums of independent random variables, its generalization for martingales, i.e. the Burkholder-Davis-Gundy inequality. This last inequality is also proved for convex functions. The section about \(U\)-statistics and some theorems concerning this are added in this third edition.

The presentation of the subject is concise, mathematically rigorous and yet very intelligible and lucid. The considerations are illustrated by many examples. There are also exercises after every section and references after each chapter. The book should be easily adapted and very useful for graduate students. It may serve also as a teaching aid for lecturers at universities and a source of reference for researchers.

Reviewer: F.Weisz (Budapest)

##### MSC:

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

60G07 | General theory of stochastic processes |

60A10 | Probabilistic measure theory |

60G40 | Stopping times; optimal stopping problems; gambling theory |

60G42 | Martingales with discrete parameter |

60F05 | Central limit and other weak theorems |