Statistical Independence in Probability, Analysis and Number Theory

Statistical Independence in Probability, Analysis and Number Theory

MARK KAC
Volume: 12
Copyright Date: 1959
Edition: 1
Pages: 109
https://www.jstor.org/stable/10.4169/j.ctt5hh8tq
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  • Book Info
    Statistical Independence in Probability, Analysis and Number Theory
    Book Description:

    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. Born in Poland, he studied with Hugo Steinhaus at Lwów earning his doctorate in 1936. He had a long and productive career in the United States, serving on the faculties of Cornell University (1939–1961), Rockefeller University (1961–1982), and the University of Southern California (1982 until his death in 1984).

    eISBN: 978-1-61444-012-3
    Subjects: Statistics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. PREFACE
    (pp. ix-xii)
    Mark Kac
  3. Table of Contents
    (pp. xiii-xiv)
  4. CHAPTER 1 FROM VIETA TO THE NOTION OF STATISTICAL INDEPENDENCE
    (pp. 1-12)

    1. A formula of Vieta. We start from simple trigonometry. Write

    $\sin x = 2\sin \frac{x}{2}\cos \frac{x}{2}$

    $ = {2^2}\sin \frac{x}{4}\cos \frac{x}{4}\cos \frac{x}{2}$

    (1.1)$ = {2^3}\sin \frac{x}{8}\cos \frac{x}{8}\cos \frac{x}{4}\cos \frac{x}{2}$

    .

    .

    .

    $ = {2^n}\sin \frac{x}{{{2^n}}}\prod\limits_{k = 1}^n {\cos \frac{x}{{{2^k}}}.} $

    From elementary calculus we know that, for$x \ne 0,$

    $1 = \mathop {\lim }\limits_{n \to \infty } \frac{{\sin \frac{x}{{{2^n}}}}}{{\frac{x}{{{2^n}}}}} = \frac{1}{x}\mathop {\lim }\limits_{n \to \infty } {2^n}\sin \frac{x}{{{2^n}}},$

    and hence

    (1.2)$\mathop {\lim }\limits_{n \to \infty } {2^n}\sin \frac{x}{{{2^n}}} = x.$

    Combining (1.2) with (1.1), we get

    (1.3)$\frac{{\sin x}}{x} = \prod\limits_{k = 1}^\infty {\cos \frac{x}{{{2^k}}}} .$

    A special case of (1.3) is of particular interest. Setting$x = \frac{\pi }{2},$we obtain

    (1.4)$\frac{2}{\pi } = \prod\limits_{n = 1}^\infty {\cos \frac{\pi }{{{2^{n + 1}}}}} $

    $ = \frac{{\sqrt 2 }}{2}\frac{{\sqrt {2 + \sqrt 2 } }}{2}\frac{{\sqrt {2 + \sqrt {2 + \sqrt 2 } } }}{2}$...,

    a classical formula due to Vieta.

    2. Another look at Vieta’s formula. So far everything has been straightforward and familiar.

    Now let us take a look a t (1.3) from a different point of view.

    It is known that every real number$t,0 \le t \le 1$, can be writtenuniquelyin the form

    (2.1)$t = \frac{{{\varepsilon _1}}}{2} + \frac{{{\varepsilon _2}}}{{{2^2}}} + ...,$

    where each ε is either 0 or 1....

  5. CHAPTER 2 BOREL AND AFTER
    (pp. 13-35)

    You have all heard that if you play a fair game of chance, then, in the long run, it is unlikely that you will get rich. “The law of averages will take care of it” is what one hears uttered wisely in this and similar connections. What is this “law of averages”? Is it some sort of a physical law, or is it a purely mathematical statement? It is mostly the latter, although the agreement with experimental evidence is remarkably good. Let us forget about experimental evidence and concentrate on the mathamatical issues. Suppose I toss a “fair” coin, winning...

  6. CHAPTER 3 THE NORMAL LAW
    (pp. 36-52)

    In § 1 of Chapter 2, we discussed the “weak law of large numbers.” A more precise result was proved by De Moivre to the effect that

    (1.1)$\mathop {\lim }\limits_{n \to \infty } \mu \{ {\omega _1}\sqrt n < {r_1}(t) + \cdots + {r_n}(t) < {\omega _2}\sqrt n \} = \frac{1}{{\sqrt {2\pi } }}\int\limits_{{\omega _1}}^{{\omega _2}} {{e^{ - {y^{1/2}}}}} dy$

    The reader will have no trouble interpreting this result in probability terms. An elementary proof can be based on formula (5.6) of Chapter 1, and (1.1) becomes equivalent to the purely combinatorial formula

    (1.2)$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{\frac{n}{2} + \frac{{{\omega _1}}}{2}\sqrt n < l < \frac{n}{2} + \frac{{{\omega _2}}}{2}\sqrt n } {\frac{1}{{{2^n}}}\left( {_l^n} \right)} = \frac{1}{{\sqrt {2\pi } }}\int\limits_{{\omega _1}}^{{\omega _2}} {{e^{ - {y^{2/2}}}}} dy.$

    Adroit use of Stirling’s formula will yield (1.2), but this proof will also obscure the nature of the theorem. Attempts to generalize (1.1) provided one of the strongest motivations for developing analytical tools of probability theory....

  7. CHAPTER 4 PRIMES PLAY A GAME OF CHANCE
    (pp. 53-79)

    A number theoretic functionf(n)is a function denned on the positive integers 1, 2, 3, · · · . The meanM{f(n)}offis denned as the limit (if it exists)

    (1.1)$M\{ f(n)\} = \mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{n = 1}^N {f(n)} $

    IfAis a set of positive integers, we denote byA(N)the number of its elements among the firstNintegers. If

    A(N) (1.2)$\mathop {\lim }\limits_{N \to \infty } \frac{{A(N)}}{N} = D\{ A\} $

    exists, it is called thedensity of A.The density is analogous to the relative measure (see § 5 of Chapter 3), and like relative measure it is not completely additive. Consider the integers divisible by a primep....

  8. CHAPTER 5 FROM KINETIC THEORY TO CONTINUED FRACTIONS
    (pp. 80-93)

    About the middle of the nineteenth century attempts were begun to unite the disciplines of mechanics and thermodynamics.

    The main problem was to derive the Second Law of Thermodynamics from the picture of matter as consisting of particles (atoms or molecules) subject to forces and obeying the laws of mechanics.

    In the hands of Maxwell and Boltzmann (and later J. W. Gibbs) this kinetic approach flowered into one of the most beautiful and far-reaching achievements of science.

    But the approach was marred, at the outset, by two paradoxes. The first, voiced in 1876 by Loschmidt, consisted in observing that the...

  9. Back Matter
    (pp. 94-94)