7. Probability and Statistics Soviet Essays - Sheynin, Oscar

charged with making the leading plenary report on the theory of probability. All thisdescribes the forms of contacts and influences of which the pre-revolutionary mathematicianswere completely deprived.Bernstein’s predecessors were almost exclusively examining sums of independent randomvariables thus continuing the traditions of the classics of probability theory 5 . However,practice poses problems that very often demand the study of series of random variables ratherconsiderably depending one on another. In most cases this dependence is the stronger thenearer in the given series are the considered variables to each other; on the contrary, variablessituated far apart occur to be independent or almost so (meteorological factors,chronologically ordered market prices). Bernstein was the first who successfully attempted togeneralize the main principles of probability theory to these cases as well. His remarkabletheorem on the law of large numbers extends its action to all series of dependent terms wherethe correlation (the measure of dependence) between the terms of the series unboundedlydecreases with an infinite increase in the distance between them. His deep investigationsdevoted to the limit theorem showed that it also, under very wide assumptions, can begeneralized to series of dependent random variables. He also was the first to formulate andsolve the problem about many-dimensional generalizations of the limit theorem which is of afundamental importance for mathematical physics (theory of diffusion, Brownian motion).Along with effectively and fittingly continuing the most glorious work done before theRevolution, the Soviet period brought about many essentially new points concerning both thesubstance of the issues under development and the forms of scientific work. Here, the mostsignificant example is the scientific school created within Moscow University, a collective ofresearchers whose like the probability theory in pre-revolutionary Russia could not haveknown. It possesses its own style, its scientific traditions, its rising generation; at the sametime, it enjoys quite a deserved reputation as one of the leading and most advanced schools inthe world. There is no important region of probability theory in whose development theMoscow school had not participated actively and influentially. It had initiated and attainedthe first achievements in a large number of modern issues whereas foreign scientists onlyjoined in their investigations later.[4] Let me attempt now to shed as much light as it is possible for an author of a paper notaddressed to specialists, on the main achievements of the Moscow school. Until the very lastyears, the theory of probability only studied infinite patterns that were sequences of randomvariables. Most often we imagined such sequences as series of consecutive values of one andthe same randomly changing (for example, with time) magnitude (consecutive readings on athermometer; consecutive positions of a particle experiencing Brownian motion). However,for suchlike patterns a direct study of a variable continuously changing with time; anexamination of a continuous interchange, of a continuous series of values where the changeoccurring between any two moments of time is subject to the action of chance, rather than ofa sequence, would better conform to reality.We thus arrive at the idea of a stochastic (a random) process where consecutiveinterchange is replaced by a continuous current conditioned by randomness. The sameproblems that occupied the theory of probability when studying sequences of randomvariables arise here, for these stochastic processes; and, in addition, a number of essentiallynew issues crop up. Along with direct theoretical interest, these stochastic processes are veryimportant for a number of applied fields (mathematical physics). However, it was difficult tocreate mathematical tools which would enable to cover them. In 1912 the French scientistBachelier had attempted to accomplish this 6 , but he did not succeed, and only in 1930Kolmogorov discovered a method based on the theory of differential equations whichensured an analytical formulation of the main problems arising in the theory of stochasticprocesses. From then onward, this theory has been actively developing and today it is one of