7. Probability and Statistics Soviet Essays - Sheynin, Oscar

where (s) are consecutive derivatives of the Gauss function and the coefficients C s (n) areexpressed through the moments of the terms k (n) . The Swedish mathematician Cramér mostfully developed this idea of Chebyshev.Any domain of mathematics having to do with determining successful approximateexpressions or with improving estimates, becomes more theoretically interesting when suchformulations of its problems are discovered that allow us to speak about best approximationsand best estimates. In the field under our consideration, such a stage of research is onlybeginning. In the foreign literature, remarkable results of this type adjoining Cramér’sfindings are due to Esseen (1945). He had studied consecutive sums n = 1 + 2 + … + n (1.3.1)of identically distributed terms 1 , 2 , …, n and Linnik [1] published a deep investigation ofthe general case of differing {pertinent} laws of distribution.1.4. Limiting Laws of Distribution for Sums of Identically Distributed Terms. Thefindings of Khinchin, Lévy and Feller on the conditions of attraction to the Gauss law(above) were obtained while studying a more general, although not less natural problemwhich is the subject of this, and of the next subsections. It consists in discovering all the lawsof distribution for sums of an increasing number of independent terms negligible ascompared with their sum.Let us begin with the simplest case of consecutive sums (1.3.1) of identically distributedterms assuming that there exist constants C n and H n > 0 such that, as n ,lim P([( n – C n )/H n ] < t) = F(t) (1.4.1)where F(t) is a non-singular law of distribution. Khinchin [39], also see Khinchin & Lévy [1],discovered the necessary and sufficient conditions which F(t) must obey to appear here as alimiting distribution. It occurred that the logarithm of the characteristic function() = ∞ −∞e it dF(t)must be expressed aslog () = i – µ|| [1 + i(/||) (; )]where , , and µ are real constants, 0 < ≤ 2, || ≤ 1, µ > 0, is arbitrary and(; ) = tg[(/2)] if 1 and = (2/)log|| otherwise.These are the so-called stable laws. If = 0 they are symmetric; Lévy considered suchlaws before Khinchin did. A linear transformation of t can lead to = 0 and µ = 1. On thecontrary, essentially differing stable laws of distribution F(t) correspond to different valuesof the parameters and . The case of = 2 is the Gauss law.Gnedenko [13] ascertained quite transparent necessary and sufficient conditions for theattraction of the sums now discussed to each of the stable laws. It is generally thought thatthe relevant limit theorems are only academic since they relate to sums of random variableswith infinite variances (finite variances lead to the Gauss law). In spite of its prevalence, thisopinion is not quite understandable because sums of independent terms with infinitevariances and even infinite expectations appear naturally indeed, for example in such a