7. Probability and Statistics Soviet Essays - Sheynin, Oscar

10. A.N. Kolmogorov. The Theory of ProbabilityIn 40 (Mathematics in the Soviet Union during 40 Years),vol. 1. Moscow, 1959, pp. 781 – 795 …Foreword by TranslatorThe references to all the essays comprising vol. 1 of the memorial publication whichincluded this essay by Kolmogorov and the essay by Gikhman & Gnedenko, also translatedin this book, were collected in vol. 2 of the same source. I extracted the bibliographypertinent for both essays just mentioned above in a Joint Bibliography appended below, afterthe second essay. In other words, my Joint Bibliography is a small portion of that vol. 2.An example is necessary. Kolmogorov cited Khinchin [133], but since he only referred to15 of the latter’s contributions, and since Gikhman & Gnedenko had not mentioned Khinchinat all, my Joint Bibliography only includes 15 Khinchin’s writings (out of the 149 listed invol. 2 of the memorial publication).[Introduction] The essay on the Soviet work in probability theory during 1917 – 1948{written by Gnedenko & Kolmogorov; translated in this book and called in the sequenceG&K} appeared in the period when the general methods of the theory of random functionsand the theory of stochastic processes with continuous time still required popularization andproof of their power and importance. Nowadays their place in science is sufficientlyascertained and the danger is rather felt of underestimating the work directed at obtainingprecise and effective results when solving concrete problems both remaining from theprevious periods of the development of probability theory and occurring in connection withnew practical requirements or in the main body of our science.Unlike G&K, this essay is being tentatively composed not chronologically (from classicalproblems to new concepts and methods) but logically (from general concepts to specialproblems, classical included). We follow the classification of the central theoretical problemsof the theory of probability itself and do not aim at explicating the use of stochastic methods.Instead, we indicate here some spheres of applications where mathematicians have beenworking systematically. These are1) Mathematical foundations of statistical physics (Khinchin [129; 131; 134; 136; 137];Yaglom [24]; Sragovich [2] and others).2) Stochastic foundations of the theory of information (Khinchin [141; 147]; Youshkevich[1]; Pinsker [3; 5]; Kolmogorov [160]; Faddeev [29]; Gelfand, Kolmogorov & Yaglom [75];Gelfand & Yaglom [79] and others).3) Queuing theory (Khinchin’s monograph [143] and others).The works of Linnik [74; 83], Kubilius [10; 14], Postnikov [10] were devoted tointramathematical applications of stochastic methods to the theory of numbers. This purelytheoretical line of research will possibly also acquire practical interest, e.g., for comparingthe real stochastic Monte-Carlo method with its number-theoretic imitations.My citing or non-citing of a certain contribution should not be considered as an attempt ofappraisal. Works of comparatively little importance can be mentioned here even at theexpense of deeper but more isolated writings if this seems opportune for illustrating thenature of some investigations of an apparently essential direction requiring widedevelopment.1. Distributions. Random Functions and Stochastic Processes