Statistical Independence in Probability, Analysis and Number Theory

by Mark Kac

Overview

This concise monograph in probability by Mark Kac, a well-known mathematician, presumes a familiarity with Lebesgue's theory of measure and integration, the elementary theory of Fourier integrals, and the rudiments of number theory. Readers may then follow Dr. Kac's attempt "to rescue statistical independence from the fate of abstract oblivion by showing how in its simplest form it arises in various contexts cutting across different mathematical disciplines."
The treatment begins with an examination of a formula of Vieta that extends to the notion of statistical independence. Subsequent chapters explore laws of large numbers and Émile Borel's concept of normal numbers; the normal law, as expressed by Abraham de Moivre and Andrey Markov's method; and number theoretic functions as well as the normal law in number theory. The final chapter ranges in scope from kinetic theory to continued fractions. All five chapters are enhanced by problems.

Product Details

ISBN-13:	9780486833408
Publisher:	Dover Publications
Publication date:	08/15/2018
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	112
File size:	15 MB
Note:	This product may take a few minutes to download.

About the Author

Mark Kac (1914–1984) was born in Poland and came to the United States in the 1930s. He taught at Cornell and later served on the faculties of Rockefeller University in New York and the University of Southern California. His main focus was probability theory, and Dover also publishes his Mathematics and Logic, co-written with S. M. Ulam.

Read an Excerpt

PREFACE

At the meeting of the Mathematical Association of America held in the Summer of 1995, I had the privilege of delivering the Hendrick Lectures. I was highly gratified when, sometime later, Professor T. Rado, on behald of the Committee on Carus Monographs, kindly invited me to expand my lectures into a monograph.

At about the same time I was honored by an invitation from Haverford College to deliver a series of lectures under the Philips Visitors Program. This invitation gave me an opportunity to try out the projected monograph on a "live" audience, and this books is a slightly revised version of my lectures delivered at Haverford College during the Spring Term of 1958.

My principal aim in the original Hedrick Lectures, as well as in this enlarged version, was to show that (a) extremely simple observations are often the starting point of rich and fruitful theories and (b) many seemingly unrelated developments are in reality variations on the same simple theme.

Except for the last chapter where I deal with a spectacular application of the ergodic theorem to continued fractions, the book is concerned with the notion of statistical independence.

This notion originated in probability theory and for a long time was handled with vagueness which bred suspicion as to its being a bona fide mathematical notion. We now know how to define statistical independence in most general and abstract terms. But the modern trend toward generality and abstraction tended not only to submerge the simplicity of the underlying idea but also to obscure the possibility of applying probabilistic ideas outside the field of probability theory.

In the pages that follow, I have tried to rescue statistical independence from the fate of abstract oblivion by showing how in its simplest form it arises in various contexts cutting across different mathematical disciplines.

As to the preparation of the reader, I assume his familiarity with Lebesgue's theory of measure and integration, elementary theory of Fourier integrals, and rudiments of number theory. Because I do not want to assume much more and in order not to encumber the narrative by too many technical details I have left out proofs of some statements.

I apologize for these omissions and hope that the reader will become sufficiently interested in the subject to fill these gaps by himself. I have appended a small bibliography which makes no pretence at completeness.

Throughout the book I have also put in a number of problems. These problems are mostly quite difficult, and the reader should not feel discouraged if he cannot solve them without considerable effort.

I wish to thank Professor C. O. Oakley and R. J. Wisner of Haverford College for their splendid cooperation and for turning the chore of traveling from Ithaca to Haverford into a real pleasure.

I was fortunate in having as members of my audience Professor H. Rademacher of the University of Pennsylvania and Professor John Oxtoby of Bryn Mawr College. Their criticism, suggestions, and constant encouragement have been truly invaluable, and my debt to them is great.

My Cornell colleagues, Professors H. Widom and M. Schreiber, have read the manuscript and arc responsible for a good many changes and improvements. It is a pleasure to thank them for their help.

My thanks go also to the Haverford and Bryn Mawr undergraduates, who were the "guinea pigs," and especially to J. Reill who compiled the bibliography and proofread the manuscript.

Last but not least, I wish to thank Mrs. Axelsson of Haverford College and Miss Martin of the Cornell Mathematics Department for the of ten impossible task of typing the manuscript from my nearly illegible notes.

Mark Kac

Ithaca, New York September, 1959

(Continues…)

---

Excerpted from "Statistical Independence in Probability, Analysis & Number Theory"
by .
Copyright © 2018 Dover Publications, Inc..
Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

---

Table of Contents

Contents Chapter 1. From Vieta to the Notion of Statistical IndependenceChapter 2. Borel and AfterChapter 3.The Normal LawChapter 4. Primes Play a Game of ChanceChapter 5. From Kinetic Theory to Continued Fractions