Statistical independence in probability analysis number theory 1st edition by Mark Kac 0486833408 9780486833408

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ISBN-10 :  0486833408 

ISBN-13 :  9780486833408

Author:  Mark Kac 

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.

Statistical independence in probability analysis number theory 1st Table of contents:

1. From Vieta to the Notion of Statistical Independence
1. A formula of Vieta
2. Another look at Vieta’s formula
3. An accident or a beginning of something deeper?
4. (½)n = ½ . . . ½ (n time)
5. Heads or tails
6. Independence and “Independence”
Problems
2. Borel and After
1. “Laws of large numbers”
2. Borel and “normal numbers”
Problems
3. “Heads or Tails”—a more abstract formulation
4. What price abstraction?
5. Example 1. Convergence of series with random signs
6. Example 2. Divergence of series with random signs
Problems
Bibliography
3. The Normal Law
1. De Moivre
2. The idea
3. Markoff’s method made rigorous
Problems
4. A closer look at the method
Problems
5. A law of nature or a mathematical theorem?
Problems
Bibliography
4. Primes Play A Game of Chance
1. Number theoretic functions, density, independence
2. The statistics of the Euler ϕ-function
Problems
3. Another application
4. Almost every integer m has approximately log log m prime divisors
Problems
5. The normal law in number theory
Problems
Bibliography
5. From Kinetic Theory to Continued Fractions
1. Paradoxes of kinetic theory
2. Preliminaries
3. Boltzmann’s reply
4. The abstract formulation
5. The ergodic theorem and continued fractions

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