7. Probability and Statistics Soviet Essays - Sheynin, Oscar

problem remained unsolved until Markov [1] showed how it was possible to obtain limitingdistributions of some of the Pearsonian types by considering an urn pattern of dependenttrials (that of an added ball). […] Pólya (1930), who apparently did not know Markov’sfindings, minutely studied this scheme of contagion, as he called it. Bernstein [34],Savkevich [1] and Shepelevsky [2] considered some of its generalizations.Kolmogorov [32] outlined another approach to theoretically justifying the Pearsoniancurves. He obtained their different types as stationary distributions that set in after a longtime in a temporal stochastic process under some assumptions about the mean velocity andvariance of the alteration of the evolving system’s random parameter. Bernstein [27] provedthat under certain conditions such a stationary distribution exists. Ambartsumian [1; 2]investigated in detail particular cases of stochastic processes leading to the main Pearsoniancurves.Romanovsky [20] generalized the Pearsonian curves to orthogonal series similar to thewell-known Gram – Charlier series, Bernstein [12; 41] also studied another stochastic patternadmitting in may practically important cases a very concrete interpretation and leading tosome transformations of the normal distribution.Still more considerable are the achievements of Soviet mathematicians in the domain ofcorrelation theory which already has vast practical applications. The works of Bernstein [13]and Khinchin [8] on the limit theorems for sums of random variables ensured a solidtheoretical foundation for the theory of normal correlation. Bernstein [41] discoveredinteresting applications of these propositions to the case of hereditary transmission ofpolymeric indications (depending on a large number of genes). His work led to a theoreticalexplanation of the law of hereditary regression empirically established by Galton.Bernstein’s research [15; 16] into the geometric foundations of correlation theory are ofparamount importance. He classified various surfaces of correlation according to simplegeometric principles. If the change of one of the random variables only results in a translationof the conditional law of the distribution (of the density), the correlation is called firm{French original: dure}. Normal correlation is obviously firm, and a firm and perfectcorrelation is always normal. If all the conditional laws of one variable corresponding tovarious values of the other one can be obtained by contracting (or expanding) one and thesame curve, the correlation is elastic. A more general type of isogeneous correlation is suchthat the elastic deformation of the conditional law is at the same time accompanied by atranslation. Bernstein derived a differential equation that enabled him to determine all thetypes of the firm correlation and some particular cases of the isogeneous type. Sarmanov [1;2] definitively completed this extremely elegant theory. The surfaces of isogeneouscorrelation are represented asF(x; y) = [Dx 2 y 2 + 2Gx 2 y + 2Exy 2 + Ax 2 + By 2 + 2Hxy + I] c .In some cases the conditional laws are expressed by the Pearson curves. In the general caseisogeneous correlation is heteroscedastic (with a variable conditional variance). Theregression curve of y on x has equationy = –GxDx22+ Hx + I+ Ex + Fand a similar equation exists for the regression of x on y.Obukhov [1; 2] developed the theory of correlation for random vectors first considered byHotelling (1936). It is widely applied in meteorological and geophysical problems, in thetheory of turbulent currents and in other fields. Making use of tensor methods, he was thefirst to offer an exposition in an invariant form. He introduced tensors of regression and of