7. Probability and Statistics Soviet Essays - Sheynin, Oscar

His (Theory of Probability), which already became classical, soonfollowed. At the same time, from 1923 -1925 onward, the works of the Moscow school(Khinchin, Kolmogorov) started developing. In the beginning, they were restricted to a rathernarrow sphere of issues within the reach of the methods taken over from the theory offunctions of a real variable 3 .1.The central issue of research done by Chebyshev, Markov and Lipaunov was theascertaining of the conditions for the applicability of the normal, or Gaussian law ofdistribution to sumss n = x 1 + x 2 + … + x n (1)of a large number of independent or weakly dependent random terms x i . Bernstein, in afundamental memoir of 1926, completed the classical methods of studying this issue. Heessentially widened these conditions for dependent terms and was the first who rigorouslyjustified the application of the many-dimensional Gaussian law to sums of vectors in spacesof any number of dimensions. This last-mentioned result also theoretically substantiates theapplicability of the formulas of normal correlation for the case in which correlated variablesmay be considered as sums of a very large number of terms with the connection betweenthese variables being restricted by that between the corresponding, or those close to them,terms of these sums. Such, in particular, is the situation when quantitative indications, causedby an additive action of a very large number of genes, are inherited. Consequently, Bernsteinwas able to show that the Galton law of the inheritance of such indications was a corollary ofthe Mendelian laws (under the assumption that a large number of uncoupled genes actadditively) and does not contradict them at all as it had been often stated.Returning to the one-dimensional case, we may formulate the conditions for theapplicability of the normal law to sums of independent terms in the following way: Withprobability close to 1, all the terms are much less than their sum s n 4 . A natural question hereis, What limit distributions can be obtained if we only require the same for each separateterm x i (the principle of individual negligibility)? G.M. Bavli and Khinchin have recentlyanswered this question (as also did P. Lévy in France in a somewhat vague form). In the limit,we obtain the so-called infinitely divisible laws that include as particular cases the Gauss, thePoisson, and the Caushy laws. These undoubtedly deserve to be more systematicallyintroduced into statistical practice as well.In addition to this essential extension of the classical approach to the limit theoremsconcerning sums of random terms, we ought to indicate the following. In applications, werefer to the limit theorems when dealing with the distributions of finite sums. At the sametime, however, the existing estimates are such that in many of the most important practicalcases the guaranteed estimate of the remainder term exceeds the main term many times over 5 .Actually, quite a satisfactory estimate only exists for the simplest case of the Laplace –Bernstein theorem.Another, even more venerable subject of classical investigations in the theory ofprobability, is the issue about the conditions for the applicability of the law of large numbersto sums of independent terms. In a number of studies, Khinchin, Kolmogorov and others alsodeveloped it further. In addition to obtaining necessary and sufficient conditions, newconcepts of strong and relative stability of the sums were created and conditions for theirapplication were formulated. Finally, Khinchin discovered an absolutely new remarkableasymptotic formula for the order of the maximal deviations of consecutive sums (1) from themean, – the so-called law of the iterated logarithm, – for sequences x 1 , x 2 ,… , x n … ofindependent terms x i . Bernstein and Khinchin studied the conditions for the applicability ofthe law of large numbers to sums of dependent terms.