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Probability essentials. (English) Zbl 0968.60003
Universitext. Berlin: Springer. x, 250 p. (2000).
This book is based on lectures held by both authors through many years. It is condensed to the “essentials” of probability theory and founded on measure theory which can be lectured in a one-semester course. The two topics are developed parallel, where the probabilistic line is the leading one, and where only the necessary tools from measure theory are presented. The authors have found a way to give a relatively short but highly concentrated course on probability theory on a mathematical level that remains understandable not only for mathematicians but also for students and scientists from engineering, economics and other fields. On the other hand, it is mathematically deep enough to open the door to a study of advanced applications of probability theory in other sciences, which use e.g. stochastic processes, martingales, Brownian motion, Itô calculus and so on.
The content follows the usual line first: From discrete to general probability spaces, in particular on \(R_1\), \(R_n\), random variables and their distributions, moments, characteristic functions, convergence of random variables, laws of large numbers, central limit theorems, \(L^2\)-theory. Then it provides an introduction to conditional expectations, martingale theory and the Radon-Nikodým theorem, which meet with increasing interest in applications of probability theory. Many carefully chosen exercises following each of the 28 chapters strongly support the reader in understanding the matter. The book can be highly recommended to students as well as to lecturers.

MSC:
60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60E05 Probability distributions: general theory
60E10 Characteristic functions; other transforms
60G42 Martingales with discrete parameter
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