7. Probability and Statistics Soviet Essays - Sheynin, Oscar

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In Kiev, in the 1930s, N.M. Krylov and N.N. Bogoliubov began their study of ergodictheorems for Markov chains. They issued from the theory of dynamic systems and thedirect cause of their research was the desire to justify the ergodic hypothesis formulatedalready by Boltzmann. Later, their problems widened and adjoined the work of theMoscow mathematicians.[4] After the Great Patriotic War {1941 – 1945} Linnik in Leningrad and his students inVilnius built up new powerful collectives working in various directions of the theory ofprobability and mathematical statistics. During the Soviet years a large number ofmonographs and textbooks in probability theory were published. Some of them woninternational recognition, have went and are going through many editions abroad 7 .Collected translations of papers of Soviet authors on probability theory and mathematicalstatistics regularly appear in the USA. The periodical % (Theory of Probability and Its Applications) is being translated and publishedthere in its entirety. In mathematical statistics, the books Smirnov & Dunin-Barkovsky(1955; 1959) 8 and Linnik (1966) should be mentioned. Bolshev & Smirnov (1968)compiled excellent tables, and a number of tables were due to Slutsky, Smirnov and others.The specialized periodical mentioned above was established owing to the considerableincrease in the amount of investigations in probability theory and mathematical statistics.In addition, writings on probability continue to appear in general mathematical and variousspecial editions. During the latest years, the number of articles devoted to the theory ofprobability and published in engineering journals greatly increased which undoubtedlytestifies that the demands of modern technology and theoretical studies accomplished bymathematicians are connected with each other and that the engineers’ level of stochasticeducation has risen.[5] The contribution of Soviet scientists to the development of the theory of probabilitydeserves highest appraisal. By creating an axiomatics of the theory on the basis of thetheories of measure and of functions of a real variable, they fundamentally transformed thescience and laid a robust foundation for the development of its new branches. The firstworks initiating the creation of the theory of stationary processes and random functions ledto the focus of scientific attention in probability theory shifting to this very field. Inaddition, owing to the development of general methods of the theory of random processesand fields, great possibilities for studying phenomena in nature and economics as well astechnological operations have opened up. Already the first steps in this direction yieldedappreciable results. The classical issue of summing independent random variables wasessentially promoted, and, in some aspects, settled. At the same time, new problemsimparted freshness and fascination to this venerable subject.The intensive progress in technology and physics advanced many unexpected problemsand initiated absolutely new directions of research, in the first place information theory(that originated in the USA), the theory of stochastic automatic machines, theory ofoptimal control of stochastic processes, reliability theory. Soviet mathematicians haveseriously contributed to the development of these new domains as well.Notes1. {Only the Laplacean theory of errors.}2. {The method of moments is due to Bienaymé and Chebyshev, and Gnedenko himselfsaid so later (Gnedenko & Sheynin 1978, p. 262).}3. {I believe that the theory of errors is the statistical method as applied to the treatmentof observations, and (unlike Gnedenko) that it had been essentially completed by the end
of the 19 th century. True, the spread of triangulation over great territories as well as thenew technology demanded the solution of many practical problems.}4. {Markov’s textbook does not make easy reading. Recall his own words (Ltter toChuprov of 1910; Ondar 1977, p. 21): I have often heard that my presentation is notsufficiently clear.}5. {Gnedenko had greatly enhanced his appraisal of Slutsky by inserting his portrait, theonly one except for Kolmogorov’s.}6. {Before ca. 1989 no academy of sciences ever existed in Soviet Russia, by far thelargest union republic.}7. {Gnedenko mentioned books of several eminent mathematicians (Kolmogorov,Khinchin, Linnik, Skorokhod, Dynkin and himself). Since then, many books, notablyKolmogorov’s Selected Works, have appeared in translation abroad. One of these was thebook mentioned in [7].}8. {The authors of the latter book were mentioned on the title-page in the opposite order,see References.}References1. Bernstein, S.N. (1911), (Theory of Probability). Kharkov.Mimeographed edition. Four ordinary editions (1927 – 1946).2. --- (1912, in French), Demonstration of the Weierstrass theorem based on the theoryof probability. Russian translation: (Coll. Works), vol. 1. N.p., 1952,pp. 105 – 107. English translation: Math. Scientist, vol. 29, 2004, pp. 127 – 128.3. Bolshev, L.N., Smirnov, N.V. (1968), ) (Tables of Mathematical Statistics). M.4. Buniakovsky, V.Ya. (1846), ! (Principles of the Mathematical Theory of Probability). Petersburg.5. Dunin-Barkovsky, I.V., Smirnov, N.V. (1955), ( (Theory of Probability and MathematicalStatistics in Technical Sciences). M.6. Ermakov, V.P. (1878; 1879), 1 (Elementary Course in the Theory of probability). Kiev.7. Gnedenko, B.V., Sheynin, O. (1978, in Russian), Theory of probability, this being achapter in Mathematics of the 19 th Century, vol. 1. Editors, A.N. Kolmogorov. A.P.Youshkevich. Basel, 1992 and 2001, pp. 211 – 288.8. Kolmogorov, A.N. (1948, in Russian), Slutsky, an obituary. Math. Scientist, vol. 27,2002, pp. 67 – 74.9. Linnik, Yu.V. (1966), '. %(Statistical problems with Nuisance Parameters). M.10. Markov, A.A. (1900), (Calculus of Probability). LaterRussian editions: 1908, 1913, 1924. German translation: 1912.11. Ondar, Kh.O., Editor (1977, in Russian), Correspondence between A.A. Markov andA.A. Chuprov. New York, 1981.12. Slutsky, E.E. (1912), ) 5 ( % (Theory of Correlation and the Elements of the Theory of DistributionCurves). Kiev.13. Smirnov, N.V., Dunin-Barkovsky, I.V. (1959), in Russian). Mathematische Statistikin der Technik. Berlin, 1969, 1973.14. Tikhomandritsky, M.A. (1898), # (Course in Theory ofProbability Theory). Kharkov.
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[3] I bear in mind the well-known p
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successes of physical statistics. B
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classes of independent facts whose
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distribution is a corollary of the
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examine in the first place the curv
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12. According to Bortkiewicz’ ter
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generality, the similarities taking
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one on another, as well as the corr
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is inapplicable because the right s
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Instead, Slutsky introduced new not
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abandoned in August 1936, but it is
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last decades, mathematicians more o
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charged with making the leading ple
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motion and a number of others) are
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phenomena. It is self-evident that
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Such new demands were formulated in
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The addition of independent random
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automatic lathes, etc. Here, the ma
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11. Kolmogorov, A.N. Grundbegriffe
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period 1 and remained, until the ap
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of the analytical tool rather than
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with probability approaching unity,
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logic. The ensuing vagueness in his
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2. Gnedenko, B.V. (1949), On Lobach
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will be sufficient, although not ne
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nlimk = 1P(| k (n) - m k (n) | > H
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favorite classical issue as the gam
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and some quite definite (not depend
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influenced by a construction that a
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P ij (1) = p ij (1) , P ij (t) =kP
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4.2d. Bebutov [1; 2] as well as Kry
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are yet no limit theorems correspon
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described, from the viewpoint that
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conditional variance and determined
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Romanovsky [45] and Kolmogorov [46]
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Let S be the general population wit
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Part 1. Russian/Soviet AuthorsAmbar
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2. On necessary and sufficient cond
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Page 86 and 87: Obukhov, A.M. 1. Normal correlation
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Page 92 and 93: 10. A.N. Kolmogorov. The Theory of
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Page 102 and 103: as it is usual for the modern theor
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Page 116 and 117: Khinchin, A.Ya. 43. Math. Ann. 101,
Page 118 and 119: 7. DAN 115, 1957, 49 - 52.Pinsker,
Page 120 and 121: Anderson, T.W., Darling, D.A. (1952
Page 122 and 123: Statistical problems in radio engin
Page 124 and 125: observations for its power with reg
Page 126 and 127: securing against mistakes (A.N. Kry
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7. Probability and Statistics Soviet Essays - Sheynin, Oscar

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