7. Probability and Statistics Soviet Essays - Sheynin, Oscar

Let E be some linear space with elements x, and P(A) – the probability that element xbelongs to set A. We denote linear functionals of x by f. Then a characteristic function ofdistribution P(A) is a function of the functional(f) = E xe ifx P(dE x ).If (f) can be expanded into a Taylor series(f) = 1 + h 1 (f 1 ) + h 2 (f 1 ; f 2 ) + h 3 (f 1 ; f 2 ; f 3 ) + …,the multilinear forms h n (f 1 ; f 2 ; …; f n ) provide the moments of the distribution P(A). Until now,only a very small number of corollaries were deduced from the indicated definitions.However, they {definitions or corollaries?} promise to be interesting, in particular for anumber of physical problems.3c. The Fokker – Planck differential equation and its conjugate underpin the derivation ofa number of new limit theorems concerning sums of a large number of random variables.Several methods for passing from integral equations determining the laws of distribution offinite sums to the corresponding differential equations were offered. The most elegant, as itseems to me, is the Petrovsky method which is a modification of the Perron method of upperand lower functions. However, it was Bernstein who obtained the fullest results in somedirections by applying another method for passing to the Fokker –Planck equations.Another series of limit theorems can be connected with the equations of step-wisestochastic processes (§3a). Whereas the first series of these propositions generalizes theLaplace – Liapunov theorem, the second one does the same with respect to the Poisson limittheorem. The theorems of the second type were gotten by Khinchin and Bavli.Introductory Literature1. Khinchin, A. (1933), Asymptotische Gesetze der Wahrscheinlichkeits-rechnung. Berlin.Note 1. {Fisher’s Statistical Methods for Research Workers first appeared in 1925.}4. A.Ya. Khinchin. The Theory of Probability in Pre-Revolutionary Russia and in theSoviet UnionFront Nauki i Tekhniki, No. 7, 1937, pp. 36 – 46Foreword by TranslatorKhinchin described the findings of the Moscow school of the theory of probability andargued that Soviet mathematics was far more advanced in these years (about 1937) thanbefore 1917. And he attempted to show that this fact was due to the favorable atmospherecreated in the Soviet Union for scientists. Khinchin’s high standing is the only reason whysuch rubbish deserved to be translated. If proof is needed, see Kolman (1931) and Sheynin(1998). Kolman (a minor mathematician and a high-ranking Parteigenosse who much laterescaped to the West) named Vernadsky, Sergei Vavilov, Ramsin et al and mathematiciansSchmidt, Kagan and Egorov as the bad guys; Egorov, for example, was indeed declared asaboteur, exiled and died soon afterwards. The horrible situation existing then (and bothearlier and later) in the Soviet Union is now widely known so that either Khinchin was forcedto state nonsense or he had been completely blind. True, the Luzin case was suddenly