7. Probability and Statistics Soviet Essays - Sheynin, Oscar

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favorite classical issue as the gambler’s ruin when considering a series of games each ofthem being continued until reaching a given loss. Kolmogorov & Sevastianov [1] indicatedmore topical applications.Let us go over now to a more general pattern of sums (1.1.6) where the terms of one andthe same sum are identically distributed but can possess different distributions for differentsums. Khinchin [41] proved that here the limiting laws F(t) that can appear in relation (1.4.1)are infinitely divisible; that is, they can be expressed as laws of distribution of a sum of anynumber of independent and identically distributed terms. Such laws of distribution appearedin connection with studying random processes with continuous time (Finetti). Kolmogorov[14; 15] discovered the general expression for infinitely divisible laws with a finite variance:the logarithms of their characteristic functions are written down aslog () = i + µ ∞ −∞S(; x) dg(x),S(; x) = [(e ix – 1 – ix)/x 2 ] if x 0 and = – 2 /2 otherwise.Here, and µ are constants and g(x) is a supplementary distribution function. The Gauss lawemerges if g(x) is reduced to the unity function E(x) = 0 at x ≤ 0 and = 1 otherwise. LaterLévy discovered the general formula for the log () of an infinitely divisible distributionwithout demanding the finiteness of the variance.1.5. Limit Theorems for Sums of Terms of Limiting Negligibility. In 1936 Bavli [2]ascertained that in case of finite variances the limiting law of distribution for sums ofarbitrarily (not necessarily identically) distributed independent terms of limiting negligibilityis always infinitely divisible. A year later Khinchin [41] proved this proposition abandoningthe demand of finiteness of the variances. This general proposition includes the finding(§1.4) relating to the sums of terms identically distributed within each sum since here thelimiting negligibility of the terms is inevitable.Gnedenko [10] exhaustively studied the conditions for attraction to infinitely divisiblelaws. The essence of his method consists in that a quite definite accompanying infinitelydivisible law of distribution is constructed for each sum of independent terms. The limitingnegligibility of the terms leads to the true law of probability coming closer to theaccompanying law. The problem about the existence of limiting laws is therefore reduced toa simpler problem concerning the existence of limiting expressions for the accompanyinglaws. From the viewpoint of applications, all this concept would have deserved to be studiedanew with regard to effectively estimating the closeness of the laws of distribution of thesums to some infinitely divisible law and discovering methods of determining the best (evenif only in the asymptotic sense) approximation of the true law of distribution of the sums byan infinitely divisible law. Such work did not even start yet.1.6. New Problems about the Limiting Behavior of the Sums. The conditions for theapplicability of the law of large numbers; the discovery of the possible types of limiting lawsof distribution for sums of a large number of terms; and of the conditions for attraction toeach of them, – all these problems inevitably appeared out of the desire for carrying theworks of Chebyshev, Markov and Liapunov to its logical conclusion.At the same time, much attention is recently being paid to new problems on the limitingbehavior of sums of a large number of independent terms. Those new problems that onlyinvolve the laws of distribution of the sums are not too peculiar and their solution demandsnot more than a small modification of the methods considered above. Such, for example, is
the problem concerning the conditions for the relative stability of sums of positive terms(Khinchin; Bobrov [1]; Gnedenko & Raikov [3]).Much more peculiar are the problems connected with estimation of the probabilities ofevents depending on the values taken by several consecutive sums [1.3.1] of one and thesame series of their independent terms. It was Borel who first drew the attention ofmathematicians to these problems which we shall consider in the sequel. Borel formulatedthe question on the conditions for the applicability of the strong law of large numbers, i.e., ofthe relationP [lim(| n – E n |/n) = 0] = 1 as n . (1.6.1)He also posed the problem of determining the asymptotic estimate of the range of oscillationsof the deviations | n – E n | which led Khinchin to his celebrated law of the iterated logarithm(below).The methods of solving such problems can be separated into two groups. The first group ofelementary methods is based on applying inequalities similar to the {Bienaymé –}Chebyshev inequality but enabling to estimate the behavior of some finite sequence of sumsrather than of one sum. In essence, already Khinchin made use of such methods in his firstwork [2] on the law of the iterated logarithm. However, it was an absolutely elementaryinequality which Kolmogorov [2] established in 1929 that was later applied most of all. Inour notation, it can be written down asP(sup| n – E n | ≥ a) ≤ var n /a 2 , 1 ≤ k ≤ n. (1.6.2)Bernstein [30; 41] remarkably strengthened it.The other group of methods should have been based on asymptotic formulas for the probabilities ofsome behavior of a long series of consecutive sums, small with regard to the sums of the terms.Kolmogorov [13; 18] offered such formulas in 1931. His method of stochastic differential equationsis also applicable to sums of certain dependent variables, and we shall therefore consider it below.Let us go now to the findings obtained while solving problems of the described type. Khinchin &Kolmogorov [1] exhaustively determined the conditions for the series [1.3.1] with n ofindependent random variables to converge with probability 1. The results concerning the strong law¡of large numbers are less full. All that can be elicited from considering the variances of the terms isKolmogorov’s sufficient (but not necessary) condition [8] about the convergence of the series(b 1 /1 2 ) + (b 2 /2 2 ) + … + (b n /n 2 ) + …It is natural to say that, similar to the case of the usual law of large numbers, the demand (1.6.1)determines the strong normal stability of the arithmetic means n /n, and to call these meansstrongly stable if there exist such constants C n thatP[lim |( n /n) – C n | = 0] = 1 as n¡ . (1.6.3)Kolmogorov [24] ascertained that for identically distributed n the existence of finite expectationswas necessary and sufficient for strong stability which is here always normal. Bobrov [4] discoveredinteresting results in this direction, but the problem of determining the necessary and sufficientconditions for strong stability remains open.1.7. The Law of the Iterated Logarithm and Related Problems. Let us consider a sequence ofrandom variables 1 , 2 , ..., n , … (1.7.1)
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[3] I bear in mind the well-known p
Page 6 and 7: successes of physical statistics. B
Page 8 and 9: classes of independent facts whose
Page 10 and 11: distribution is a corollary of the
Page 12 and 13: examine in the first place the curv
Page 14 and 15: 12. According to Bortkiewicz’ ter
Page 16 and 17: generality, the similarities taking
Page 18 and 19: one on another, as well as the corr
Page 20 and 21: is inapplicable because the right s
Page 22 and 23: 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 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 74 and 75: Let S be the general population wit
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
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Page 102 and 103: as it is usual for the modern theor
Page 104 and 105: 1. {The second reference to Pugache
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Smirnov, Romanovsky and others made
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determined the precise asymptotic c
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for finite values of N, M and R 2 .
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Mikhalevich’s findings by far exc
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Uch. Zap. = Uchenye ZapiskiUkr = Uk
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Khinchin, A.Ya. 43. Math. Ann. 101,
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7. DAN 115, 1957, 49 - 52.Pinsker,
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Anderson, T.W., Darling, D.A. (1952
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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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