Radically Elementary Probability Theory. (AM-117), Volume 117 (Annals of Mathematics Studies) by Edward Nelson (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Nelson develops a new approach to probability theorythat is just as powerful as but much simpler than conventional”Kolmogorov-style” probability theory used throughout mathematicsfor most of the 20th century. This book has two “radical” innovations.The first innovation is a very simple version of nonstandard analysis,much simpler than Abraham Robinson’s original version, developed in only eightpages, just powerful enough to define the arithmetic of infinitesimaland unlimited real numbers.The second innovation is a very simple version of probability theorythat restricts all sample spaces to be finite (though perhaps unlimited)and all probabilities of outcomes to be nonzero (though perhaps infinitesimal).Thus there is no need for measure theory or any other PhD level mathematics.As Nelson says in his preface, “the mathematical background required is littlemore than that which is taught in high school, and it is my hope that it willmake deep results from the modern theory of stochastic processes readilyavailable to anyone who can add, multiply, and reason.” The levelof sophistication required of readers is that of undergraduate math majors.The reduction of technical difficulty can be seen by his going from nothing to”a version of the de Moivre-Laplace central limit theorem thatcontains Lindeberg’s theorem on the sufficiency of his condition, Feller’stheorem on its necessity, Wiener’s theorem on the continuity of thetrajectories of his process, the Levy-Doob characterization if it as theonly normalized martingale with continuous trajectories, and the invarianceprinciple of Erdos and Kac as extended by Donsker and Prohorov” in just 79pages.This book will be of no use to anyone who just wants to learn conventionalprobability theory. But it is essential for anyone who wants aperspective from outside conventional theory. Anyone brainwashed bya PhD level probability course has a hard time seeing that probabilitytheory can be any other way (although it was very different in the 19thcentury and many different alternative approaches were tried in the 20th).Nelson’s book is an eye opener.

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