## Sur la loi forte des grandes nombres. Offprint from: Comptes rendus des séances de l’Académie des Sciences (Paris), t. 191.

Paris: Gauthier-Villars, [1930].

First edition, incredibly rare offprint, of one of Kolmogorov’s most famous and important works in probability theory. “Kolmogorov’s strong law of large numbers is certainly one of the most important results – and arguably *the* most important result – in the entire subject of probability theory” (Bingham, p. 156). “Kolmogorov's strong law is a supremely important result, as it captures in precise form the intuitive idea (the ‘law of averages’ of the man in the street) identifying probability with limiting frequency. One may regard it as the culmination of 220 years of mathematical effort, beginning with J. Bernoulli’s *Ars Conjectandi* of 1713, where the first law of large numbers (weak law for Bernoulli trials) is obtained” (Kendall, p. 54). “Kolmogorov was one of the twentieth century’s greatest mathematicians. He made fundamental contributions to probability theory, algorithmic information theory, the theory of turbulent flow, cohomology, dynamical systems theory, ergodic theory, Fourier series, and intuitionistic logic. Mathematical talent at this level of creativity and versatility is rarely encountered … Kolmogorov’s most famous contributions are to the foundations of probability theory” (DSB). “Andrei Nikolaevich Kolmogorov (Tambov 1903, Moscow 1987) was one of the most brilliant mathematicians that the world has ever known. Incredibly deep and creative, he was able to approach each subject with a completely new point of view: in a few magnificent pages, which are models of shrewdness and imagination, and which astounded his contemporaries, he changed drastically the landscape of the subject. Most mathematicians prove what they can, Kolmogorov was of those who prove what they want” (Charpentier *et al*., p. 1). “From the mid-seventeenth century, probability had been explored in a somewhat unsystematic fashion. By bringing to bear on the topic the apparatus of measure theory, Kolmogorov’s principal work in probability theory, *Grundbegriffe der Wahrscheinlichkeitsrechnung* (1933), established probability theory as a core area of rigorous mathematics. In so doing he transformed one-half of David Hilbert’s sixth problem: ‘To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probability and mechanics’ … The book was the culmination of an interest in probability that had begun as a collaboration with Aleksandr Y. Khinchin in 1924. This led in the ensuing four years to Kolmogorov’s publishing his celebrated three-series theorem (1928), which gives necessary and sufficient conditions for the convergence of sums of independent random variables, to his discovering necessary and sufficient conditions for the strong law of large numbers (1930) [the offered paper], and to his proving the law of the iterated logarithm for sums of independent random variables (1929)” (*ibid*.). We have been unable to locate any other copy of this offprint: no copy is listed on OCLC, KVK, etc., and there is no copy in auction records.

“The prehistory of the law of large numbers, like that of probability itself, is obscure and debatable. Hints of the idea of probability as linked with limiting frequency can be found in the written record as early as the work of Cardano (1501-1576), in his book *De Ludo Aleae* (written in 1526, and published posthumously in 1663, after the Pascal – Fermat correspondence of 1654). ‘There can be no doubt that he had a fairly good idea of the rule which is now called the law of large numbers’ (Ore, *Cardano. The Gambling Scholar* (1953), p. 120). However, as both games of chance and the basics of combinatorics go back to antiquity, it is surprising that such ideas did not emerge much earlier …

“The first theorem recognisable as a precise form of a limiting-frequency statement (or ‘law of large numbers’ in the terminology introduced later by Poisson) is the famous ‘weak law of large numbers for Bernoulli trials’ of James Bernoulli (1654-1705): if *S _{n}* is the number of successes observed in

*n*independent trials with success probability

*p*, then

*S*converges to

_{n}/n*p*in probability. [This means that, for any number

*h*> 0, the probability that

*S*differs from

_{n}/n*p*by more than

*h*tends to zero as

*n*tends to infinity.] This is the most important single result in Bernoulli’s classic book

*Ars Conjectandi*, published posthumously in 1713 … One can hardly overstate the importance of Bernoulli’s theorem: Kolmogorov, writing in 1986 in the preface to a book on Bernoulli, describes work prior to

*Ars Conjectandi*as only the prehistory of probability proper, and its true history as beginning with Bernoulli’s theorem …

“Hilbert included, as part of Problem 6 of his famous problem list of 1900, an axiomatic treatment of probability. The development of measure theory by

Lebesgue around 1901-1904 created for the first time the language in which

results on convergence with probability one could be formulated and the

machinery for proving them. [A sequence of random variables *X _{n}* is said to ‘converge with probability one’ to a random variable

*X*if the probability that

*X*converges to

_{n}*X*is 1.] … It was by now necessary to distinguish ‘in probability’ and ‘almost sure’ laws of large numbers” (Bingham).

‘Weak’ laws of large numbers involve convergence in probability; when almost sure convergence is involved, we have a ‘strong law’ (the terms ‘weak’ and ‘strong’ are used because almost sure convergence implies convergence in probability, but the converse is not true). In the offered paper, we find the following strong law of large numbers. Suppose that we have independent random variables *X _{n}* all of which have the same mean, say

*μ*. Let

*var*be the variance of

_{n}*X*, i.e. the mean of (

_{n}*X*)

_{n}– μ^{2}(

*μ*and the

*var*are all assumed to be finite). Then, the sequence of averages (sometimes called the ‘sample means’)

_{n}(*X*_{1} + *X*_{2} + … + *X*_{n})/*n*

converges almost surely to the average *μ* if, and only if, the sum

*var*_{1} + *var*_{2}/4 + *var*_{3}/9 + …

is finite. An important special case is when the *X _{n}* all have the same probability distribution. In that case, Kolmogorov’s condition is satisfied since

*var*does not depend on

_{n}*n*and

1 + 1/4 + 1/9 + … < ∞.

An example would be the roll of a fair, six-sided die – *X _{n}*is the number shown on the

*n*th roll of the die. The

*X*in this simple situation all have the same probability distribution, so Kolmogorov’s theorem states that the probability is one that the sample means converge to the mean

_{n}(1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5.

“Kolmogorov’s mother died giving him birth; he was raised by her sister and took his maternal grandfather’s family name. His aunt moved with him to Moscow when he was seven years old, where he demonstrated an early interest in biology and history. In 1920, as yet undecided over a career, he enrolled simultaneously at Moscow State University to study history and mathematics and at the Mendeleev Chemical Engineering Institute to study metallurgy. However, he soon revealed a remarkable talent for mathematics and specialized in that subject. As a 19-year-old student he was entrusted with teaching mathematics and physics courses in the Potylikhin Experimental School, and by the time he graduated in 1925 he had published 10 mathematical papers, most of them on trigonometric series—an extraordinary output for a student. This astonishing outburst of mathematical creativity continued as a graduate student with eight more papers written through 1928. He later expanded the most important of these papers, ‘General Theory of Measure and Probability Theory’—which aimed to develop a rigorous, axiomatic foundation for probability—into an influential monograph *Grundbegriffe der Wahrscheinlichkeitsrechnung* (1933). In 1929, having completed his doctorate, Kolmogorov was elected a member of the Institute of Mathematics and Mechanics at Moscow State University, with which he remained associated for the rest of his life. In 1931, following a radical restructuring of the Moscow mathematical community, he was elected a professor. Two years later he was appointed director of the Mathematical Research Institute at the university, a position he held until 1939 and again from 1951 to 1953. In 1938 he was chosen to head the new department of probability and statistics at the Steklov Mathematical Institute of the U.S.S.R. Academy of Sciences in Moscow (now the Russian Academy of Sciences), a position that he held until 1958. He was elected to the Academy of Sciences in 1939, and between 1946 and 1949 he was also the head of the Turbulence Laboratory of the U.S.S.R. Academy of Sciences Institute of Theoretical Geophysics in Moscow.

“Of the many areas of pure and applied mathematical research to which Kolmogorov contributed, probability theory is unquestionably the most important, in terms of both the depth and breadth of his contributions. In addition to his work on the foundations of probability, he contributed profound papers on stochastic processes, especially Markov processes. In Markov processes only the present state has any bearing upon the probability of future states; states are therefore said to retain no ‘memory’ of past events. Kolmogorov invented a pair of functions to characterize the transition probabilities for a Markov process and showed that they amount to what he called an “instantaneous mean” and an “instantaneous variance.” Using these functions, he was able to write a set of partial differential equations to determine the probabilities of transition from one state to another. These equations provided an entirely new approach to the application of probability theory in physics, chemistry, civil engineering, and biology” (Britannica).

Bingham, ‘The work of А. N. Kolmogorov on strong limit theorems,’ *Teoriya Veroyatnostei i ee Primeneniya*. 34 (1989), pp. 152–164

(http://www.mathnet.ru/links/b1e57cf5d14a428496071dee4c4ba10d/tvp1240.pdf). Charpentier, Lesne & Nikolski (eds.), *Kolmogorov’s Heritage in Mathematics*, 2004. Kendall, ‘Andrei Nikolaevich Kolmogorov (1903-1987),’ *Bulletin of the London Mathematical Society* 22 (1990), pp. 31-100.

4to, pp. [2], [2, blank] (closed tear in inner margin extending slightly into printed area but not affecting text, a couple of other insignificant marginal tears). Printed on two pages of a single bifolium. A remarkable survival of a very fragile document.

Item #5877

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Price:
$5,000.00
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