**The Carus Mathematical Monographs**

Volume: 12;
1959;
94 pp;
Hardcover

**Print ISBN: 978-0-88385-025-1
Product Code: CAR/12**

List Price: $25.00

AMS Member Price: $18.75

MAA Member Price: $18.75

**Electronic ISBN: 978-1-61444-012-3
Product Code: CAR/12.E**

List Price: $25.00

AMS Member Price: $18.75

MAA Member Price: $18.75

# Statistical Independence in Probability, Analysis, and Number Theory

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*Mark Kac*

MAA Press: An Imprint of the American Mathematical Society

Professor Kac's monograph is designed to
illustrate how simple observations can be made the starting point of
rich and fruitful theories and how the same theme recurs in seemingly
unrelated disciplines. An elementary but thorough discussion of the
game of "heads or tails," including the normal law and the laws of
large numbers, is presented in a setting in which a variety of purely
analytic results appear natural and inevitable. The chapter "Primes
Play a Game of Chance" uses the same setting in dealing with problems
of the distribution of values of arithmetic functions. The final
chapter "From Kinetic Theory to Continued Fractions" deals with a
spectacular application of the ergodic theorems to continued
fractions.

Mark Kac conveyed his infectious enthusiasm for mathematics
and its applications in his lectures, papers, and books. Two of his
papers won Chauvenet awards for expository excellence.

#### Reviews & Endorsements

Terrific, friendly and profound.

-- Persi Diaconis

This is a splendid book. It ranges from the primitive idea of statistical independence to applications of the most diverse sort: coin-tossing, anharmonic oscillations, prime numbers, and continued fractions. And it does all that with Kac's customary clarity and charm. Every friend of probability should have it.

-- Henry McKeon

# Table of Contents

## Statistical Independence in Probability, Analysis, and Number Theory

- Cover cover11
- Statistical Independence in Probability, Analysis and Number Theory vii8
- Copyright Page viii9
- Contents xiii14
- Chapter 1. From Vieta to the Notion of Statistical Independence 116
- Chapter 2. Borel and After 1328
- 1. "Laws of large numbers" 1328
- 2. Borel and "normal numbers" 1530
- Problems 1833
- 3. "Heads or Tails"—a more abstract formulation 2136
- 4. What price abstraction? 2338
- 5. Example 1. Convergence of series with random signs 2439
- 6. Example 2. Divergence of series with random signs 3146
- Problems 3449
- Bibliography 3550

- Chapter 3. The Normal Law 3651
- Chapter 4. Primes Play a Game of Chance 5368
- 1. Number theoretic functions, density, independence 5368
- 2. The statistics of the Euler ф-function 5469
- Problems 6277
- 3. Another application 6479
- 4. Almost every integer m has approximately log log m prime divisors 7186
- Problems 7489
- 5. The normal law in number theory 7489
- Problems 7994
- Bibliography 7994

- Chapter 5. From Kinetic Theory to Continued Fractions 8095