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Edward Nelson, "Radically Elementary Probability Theory"

Review:

Nelson develops a new approach to probability theory

that is just as powerful as but much simpler than conventional

"Kolmogorov-style" probability theory used throughout mathematics

for 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 eight

pages, just powerful enough to define the arithmetic of infinitesimal

and unlimited real numbers.

The second innovation is a very simple version of probability theory

that 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 little

more than that which is taught in 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." The level

of 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 that

contains Lindeberg's theorem on the sufficiency of his condition, Feller's

theorem on its necessity, Wiener's theorem on the continuity of the

trajectories of his process, the Levy-Doob characterization if it as the

only normalized martingale with continuous trajectories, and the invariance

principle of Erdos and Kac as extended by Donsker and Prohorov" in just 79

pages.

This book will be of no use to anyone who just wants to learn conventional

probability theory. But it is essential for anyone who wants a

perspective from outside conventional theory. Anyone brainwashed by

a PhD level probability course has a hard time seeing that probability

theory can be any other way (although it was very different in the 19th

century and many different alternative approaches were tried in the 20th).

Nelson's book is an eye opener.

Charles J. Geyer (Minneapolis, MN USA)

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Rating:
2.5 out of 5 by Book123