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Radically elementary probability theory. (English) Zbl 0651.60001
Annals of Mathematics Studies, No. 117. Princeton, New Jersey: Princeton University Press. IX, 97 p.; Cloth: $ 40.00; Paper: $ 15.00 (1987).
As the author states, this book represents an attempt to lay new foundations for probability theory, using nonstandard analysis. The mathematical background required is little more than that which is taught at high school, and it is expected that this approach will give deep results from the modern theory of stochastic processes readily available to anyone who can add, multiply, and reason.
The book discusses very briefly the following chapters: Random variables. Algebras of random variables. Stochastic processes. External concepts. Infinitesimals. External analogues of internal notions. Properties that hold almost everywhere. \(L^ 1\)-random variables. The decomposition of a stochastic process. The total variation of a process. Convergence of martingales. Fluctuations of martingales. Discontinuities of martingales. The Lindeberg condition. The maximum of a martingale. The law of large numbers. Nearly equivalent stochastic processes. The de Moivre-Laplace- Lindeberg-Feller-Wiener-Lévy-Doob-Erdős-Kac-Donsker-Prochorov theorem.
There is also an appendix directed to those who may question either the power of the present results or the worthiness of learning nonstandard analysis.
Reviewer: A.N.Philippou

60A05 Axioms; other general questions in probability
03Hxx Nonstandard models
60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
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