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Radically Elementary Probability Theory.
Using only the very elementary framework of finite probability spaces, this book treats a number of topics in the modern theory of stochastic processes. This is made possible by using a small amount of Abraham Robinson's nonstandard analysis and not attempting to convert the results into conventional form.
107 pages, Paperback
First published August 1, 1987
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September 18, 2025
Preface, page vii: "The mathematical background required is little more than high school, and it is my hope that it will make deep results from the modern theory of stochastic processes readily available to anyone who can add, multiply, and reason."
Page 5: "The set R^(omega) of all random variables on (big omega) is not only a vector space, it is an algebra."
Page 20: "Most of the concepts of analysis are vacuous when applied to a function whose domain is a finite set of points"
Page 33: "Let P: t |-> P{subscript t} be an increasing function from T to the set of all algebras of random variables on <(big omega), pr>. This is called a filtration."
Page 75: "The following is a version of the de Moivre-Laplace central limit theorem that contains Lindberg's theorem on the sufficiency of his condition, Feller's condition on its necessity, Wiener's theorem on the continuity of trajectories for his process, the Levy-Doob characterization of it as the only normalized martingale with continuous trajectories, and the invariance principle of Erdos and Kac as extended by Donsker and Prokorov."
So look, this is a fascinating reconstruction of probability theory using nonstandard analysis instead of measure-theoretic foundations. It allows for the interconnection of spaces to random variables to stochastic processes to martingales, and finding equivalences between them. But "radically elementary" is a misnomer, this is well, well beyond "little more than high school" that it's ridiculous to say anything close to it.
Page 5: "The set R^(omega) of all random variables on (big omega) is not only a vector space, it is an algebra."
Page 20: "Most of the concepts of analysis are vacuous when applied to a function whose domain is a finite set of points"
Page 33: "Let P: t |-> P{subscript t} be an increasing function from T to the set of all algebras of random variables on <(big omega), pr>. This is called a filtration."
Page 75: "The following is a version of the de Moivre-Laplace central limit theorem that contains Lindberg's theorem on the sufficiency of his condition, Feller's condition on its necessity, Wiener's theorem on the continuity of trajectories for his process, the Levy-Doob characterization of it as the only normalized martingale with continuous trajectories, and the invariance principle of Erdos and Kac as extended by Donsker and Prokorov."
So look, this is a fascinating reconstruction of probability theory using nonstandard analysis instead of measure-theoretic foundations. It allows for the interconnection of spaces to random variables to stochastic processes to martingales, and finding equivalences between them. But "radically elementary" is a misnomer, this is well, well beyond "little more than high school" that it's ridiculous to say anything close to it.
Displaying 1 of 1 review