7. Probability and Statistics Soviet Essays - Sheynin, Oscar

generality, the similarities taking place between the main notions of the theory of probabilityand the problem of metrization. Work in this direction is continuing. Kolmogorov’s yetunpublished research promises to advance considerably our understanding of the limitingstochastic regularities.3. During these years, the estimation of the probabilities connected with an infinitelycontinued series of trials acquired both an essential theoretical importance and a considerablepractical interest. The Italian, and the Moscow mathematical schools, independently of oneanother, advanced the so-called strong law of large numbers as one of the general and mainlaws of probability theory, which Borel discovered in the simplest cases already long ago.However, whereas the Italians (Cantelli) did not go further than its formulation, we haveminutely worked out both its connection with the usual law of large numbers and theconditions for its applicability (Khinchin). We were also the first to discover the law ofiterated logarithm that determined, in a certain sense, the precise upper bound of thedeviations of sums of large numbers of random variables from their expectations (Khinchin);and to establish that it has a very wide field of applications (Kolmogorov). These issues arerecently attracting considerable attention of the European (mostly French and Italian)scientists.4. The investigation of more subtle limiting regularities under the conditions of theclassical Bernoulli pattern, and especially the study of the behavior of the distributionfunction at large distances from the center, which are also of a considerable practical interest,had mostly been developed during these years in the Soviet Union. We derived here anumber of findings exhausting the posed problems (Smirov. Khinchin).5. During the last years, after some interruption, the interest towards the simplest case of aseries of dependent random variables known as a Markov chains has again strengthened. Thefirst pertinent works had appeared abroad (Hadamard; Hostinsky), but they also met with arapid response in our country and were supplemented (Romanovsky) and considerablygeneralized (Kolmogorov). Important investigations on the applicability of the limitingtheorem to Markov chains (Bernstein) constitute a separate entity.6. Kolmogorov extended the theory of Markov chains to the continual case thus convertingit into a general theory of stochastic processes. This is one of the most remarkableachievements of Soviet mathematics in general. The new theory covers any process wherethe instantaneous state of a system uniquely determines the probability of any of its states atany subsequent moment irrespective of its previous history. Mathematically speaking, thistheory establishes for the first time the general principles connecting the problems ofstochastic processes with differential equations of definite types. Kolmogorov especiallyconsidered processes where the distribution function of the increment of the random variableremained constant, independent either of time or of the value of the variable at a givenmoment. He derived the general analytical form of such processes. Parallel investigationsabroad were only carried out in Italy (Finetti) where some particular results were attained.7. Along with Markov chains attention during the latest years was attracted to stationaryseries of random variables, that is, to series in which all the terms have the same expectationand the same variance and the correlation coeffi-cient between two terms only depends ontheir mutual location in the series.In some aspects, these series, of essential importance for various applications, present anextension of Markov chains. Soviet mathematicians have developed their theory (which,however, is yet far from being completed). In the first instance, we derived series whose