7. Probability and Statistics Soviet Essays - Sheynin, Oscar

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Let S be the general population with a discrete argument taking values x 1 , x 2 , …, x s withprobabilities P 1 , P 2 , …, P s (P i > 0 and P 1 + P 2 + … + P s = 1); let also the function = (x 1 ,x 2 , …, x s ; P 1 ; P 2 ; …; P s ) represent some characteristic of the general population and T =(x 1 , x 2 , …, x s ; m 1 /n; m 2 /n; …; m s /n) be the corresponding characteristic of the empiricaldistribution obtained by replacing the P i ’s by the frequencies m i /n of the sample values of .If is continuous with respect to all the P i ’s, then, as n , P(|T – | < ) converges to 1uniformly with respect to the P i ’s for any > 0.Sarymsakov specified this result by discovering that, as n , a stronger relation P(T )= 1 takes place.Romanovsky [19] also considered a number of problems, this time of the parametric type,being connected with testing hypotheses of whether two independent samples belonged toone and the same normal population. Of special importance is his derivation of thedistribution of the -test (also introduced by him) which found application in the so-calledanalysis of variance in agronomy and other similar fields. He [42] also studied statisticalproblems relating to series of events connected into a Markov chain, indicated methods forempirically ascertaining the law of the chain and for testing the hypothesis of its simplicity.4. Problems of Prediction. Discovering Periodicities. SomeApplications of Statistical MethodsIn concluding my not at all comprehensive review, I indicate a number of writings devotedto quite concrete problems but at the same time having a considerable general methodicalinterest. Slutsky’s studies [8; 16; 17; 18; 20; 21; 22; 24; 25] occupy a prominent place amongthese. They were devoted to connected time series and prediction and extrapolation, came tobe widely known and enjoyed considerable response in the world literature. Slutsky,Khinchin and Kolmogorov largely created the theory of continuous stochastic processes. Thisled Slutsky to extremely interesting conclusions about pseudoperiodic properties of someclasses of stationary random series (the limiting sinusoidal law). His further studies in thisfield were followed by the reconstruction of the Schuster theory of periodograms onincomparably broader foundation as well as by the revision of all the usual methods ofestimating statistical constants only valid when the consecutive observations wereindependent. Slutsky threw new light on problems of correlation, prediction andextrapolation of connected series which had already for a long time attracted the attention ofthe most prominent representatives of our national statistics (Obukhov, N.S. Chetverikov,B.S. Yastremsky). With regard to the deepness of theoretical penetration into the nature ofthese difficult problems still awaiting to be scientifically solved, and to the wit of themethods applied, his investigations leave similar attempts made by West European andAmerican statisticians far behind.Peculiar problems of congestion that occur when providing mass service (automatictelephony) were the subject of deep research accomplished by Khinchin [19; 24] andKolmogorov and continued by Volberg [1] and Bukhman [1; 2]. Obukhov [5], whilestudying the theory of turbulence by developing the profound ideas introduced there byKolmogorov, worked out very complicated problems in statistical description of continuousrandom fields. A new development occurred in the problem of the general structure of meanvalues (Kolmogorov [9], Konius [1], Boiarsky et al [1]).The penetration of statistical methods into most various fields of research is characterizedby the abundance of new problems demanding special approaches and methods for solvingthem. Here I only mention the very interesting works of Kolmogorov on the theory ofcrystallization of metals [28] and on the law of distribution of the sizes of crushed particles
[40] and on the writings of a number of hydrologists on economic calculations (S.I. Kritsky,M.F. Menkel, A.D. Savarensky et al).We can be justly satisfied in that the Soviet statistical thought had undoubtedly played afundamental part in a number of paramount problems and provided specimens not yetsurpassed with respect to deepness or ideological richness.Notes1. {With respect to this, nowadays hardly well-known rule, see Sheynin, O. (1994),Bertrand’s work on probability. Arch. Hist. Ex. Sci., vol. 48, pp. 155 – 199 (p. 190, Note57).}2. {Romanovsky [45] contained no new ideas worthy of mention. As many other mosteminent mathematicians, he had not really studied the theory of errors.}9. Joint Bibliography to the Papers by Gnedenko & Kolmogorov and SmirnovForeword by TranslatorThe Bibliography was definitely corrupted by mistakes, and the page numbers weresometimes missing. When possible, I corrected/supplemented it by consulting M.G. Kendall& Alison G. Doigt, Bibliography of Statistical Literature Pre-1940. Edinburgh, 1968. Theauthors referred to a few foreign sources in separate footnotes; I collected these and now theyconstitute a very short second part of the Bibliography. I have additionally listed thecollected works of a few scholars published after 1948; in such cases, I did not mentionlesser known periodicals in which their pertinent papers had initially appeared. I regret to addthat the Bibliography in Bogoliubov, A.N. & Matvievskaia, G.P. Romanovsky. Moscow,1997, was compiled carelessly. The DAN (see Abbreviations) were also published as the C.r.Acad. Sci. of the Soviet Union with the contributions appearing there in one of the three mainEuropean languages. If not stated (or not immediately apparent), the contributions are inRussian.AbbreviationsAN = Akademia NaukC.r. = C.r. Acad. Sci. ParisDAN = Doklady AN USSRFAN = Filial ANGIIA = G. Ist. Ital. AttuariIAN = Izvestia AN USSR. If not indicated, the series is MatematikaIMM = Inst. Matematiki i MekhanikiL = LeningradLGU = Leningradsk. Gosudarstvenny Univ.M = MoscowMGU = Moskovsky Gosudarstvenny Univ.MS = Matematich. SbornikNI = Nauchno-Issledovatelsk.SAGU = Sredneaziatsk (Central Asian) Gosudarstvenny Univ.SSR = Soviet Socialist Republic (e.g., Uzbekistan or Ukraine)U = in UkrainianUch. Zap. = Uchenye ZapiskiUMN = Uspekhi Matematich. NaukUz = UzbekVS = Vestnik Statistiki
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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
Page 24 and 25: abandoned in August 1936, but it is
Page 26 and 27: last decades, mathematicians more o
Page 28 and 29: charged with making the leading ple
Page 30 and 31: motion and a number of others) are
Page 32 and 33: phenomena. It is self-evident that
Page 34 and 35: Such new demands were formulated in
Page 36 and 37: The addition of independent random
Page 38 and 39: automatic lathes, etc. Here, the ma
Page 40 and 41: 11. Kolmogorov, A.N. Grundbegriffe
Page 42 and 43: period 1 and remained, until the ap
Page 44 and 45: of the analytical tool rather than
Page 46 and 47: with probability approaching unity,
Page 48 and 49: logic. The ensuing vagueness in his
Page 50 and 51: 2. Gnedenko, B.V. (1949), On Lobach
Page 52 and 53: will be sufficient, although not ne
Page 54 and 55: nlimk = 1P(| k (n) - m k (n) | > H
Page 56 and 57: favorite classical issue as the gam
Page 58 and 59: and some quite definite (not depend
Page 60 and 61: influenced by a construction that a
Page 62 and 63: P ij (1) = p ij (1) , P ij (t) =kP
Page 64 and 65: 4.2d. Bebutov [1; 2] as well as Kry
Page 66 and 67: are yet no limit theorems correspon
Page 68 and 69: described, from the viewpoint that
Page 70 and 71: conditional variance and determined
Page 72 and 73: Romanovsky [45] and Kolmogorov [46]
Page 76 and 77: Part 1. Russian/Soviet AuthorsAmbar
Page 78 and 79: 2. On necessary and sufficient cond
Page 80 and 81: Gnedenko, B.V., Groshev, A.V. 1. On
Page 82 and 83: 52. ( (Mathematical Principl
Page 84 and 85: Kozuliaev, P.A. 1. Sur la répartit
Page 86 and 87: Obukhov, A.M. 1. Normal correlation
Page 88 and 89: 30. Généralisations d’un théor
Page 90 and 91: 22. Alcune applicazioni dei coeffic
Page 92 and 93: 10. A.N. Kolmogorov. The Theory of
Page 94 and 95: Kuznetsov, Stratonovich & Tikhonov
Page 96 and 97: In the homogeneous case H s t = H t
Page 98 and 99: to such a generalization. He only s
Page 100 and 101: In the particular case of a charact
Page 102 and 103: as it is usual for the modern theor
Page 104 and 105: 1. {The second reference to Pugache
Page 106 and 107: Smirnov, Romanovsky and others made
Page 108 and 109: determined the precise asymptotic c
Page 110 and 111: for finite values of N, M and R 2 .
Page 112 and 113: Mikhalevich’s findings by far exc
Page 114 and 115: Uch. Zap. = Uchenye ZapiskiUkr = Uk
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
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observations for its power with reg
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securing against mistakes (A.N. Kry
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of the others, then its distributio
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In Kiev, in the 1930s, N.M. Krylov
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7. Probability and Statistics Soviet Essays - Sheynin, Oscar

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