7. Probability and Statistics Soviet Essays - Sheynin, Oscar

the most urgent chapters of probability, rich in results and problems. Along with the Moscowschool, a number of foreign mathematicians, with whom we have been in constant contact,were participating in the pertinent work.During the last decades, the theory of probability encountered extremely diverse limitingprocesses. It is about time for a mature mathematical science {such as probability} to sort outthis variety, to establish the connections and interrelations between these limiting formations.Only in 1936 the French scientist Fréchet systematically explicated this topology of randomvariables in a complete form that demanded a half of a lengthy treatise. However, as herepeatedly indicated, the founder of the theory, who had first developed it considerably, wasthe Moscow mathematician Slutsky. He also played a substantial part in working out thetheory of stochastic processes (above); he highly successively studied the application of themain analytic operations (integration, differentiation, expansion into Fourier series, etc) onrandom variables and often connected his investigations with problems arising in appliednatural sciences.One of the objects initially studied by Markov but then pretty well forgotten was the seriesof random variables now usually called Markov chains. These are sequences of mutuallydependent random variables connected by an especially simple dependence when the law ofdistribution of any of them is completely defined (in the most simple case) by the value ofthe immediately preceding variable so that the influence of the earlier history is eliminated.Markov rather considerably developed the study of such chains but after his death his resultswere forgotten whereas foreign scientists probably never heard of them at all. This happenedpartly because of the abovementioned causes, and partly because scientists in those dayswere yet unable to connect these theoretical investigations with current issues in naturalsciences or practice. In 1928, when the need to study such chains became necessary forvarious reasons, and when their possibilities for applications were clearly outlined, they werediscovered for the second time. Foreign scientists, believing that they were turning up virginsoil, proved a number of his findings anew. From then onwards, the theory of Markov chainsbecame, and is remaining one of the most intensively developing chapters of the theory ofprobability. The Moscow school, mainly in the person of Kolmogorov, actively and verysuccessively participated in this work. I ought to add right here, that one of our outstandingspecialists, Romanovsky (Tashkent), who does not belong to the Moscow school, was and isalso energetically taking part here, so that the total contribution of the Soviet science to thecreation of this theory that originated in Russia seems very considerable.Then, the Moscow school initiated and attained the main achievements in developing asphere of issues directly adjoining the classical epoch of probability theory. It was knownlong ago that in the most important instances the deviation of the arithmetic mean of a longseries of independent random variables from its expectation obey the normal law and that, inparticular, the value of such a deviation is therefore, in a sense, and to a certain extent,bounded, but until recently the problem of determining the exact boundaries for thesedeviations did not arise even for the most elementary cases. In 1924, Khinchin firstformulated and solved this problem for the Bernoulli trials; then, after he, somewhat later,had extended this solution (now known as the law of the iterated logarithm) to some moregeneral cases, Kolmogorov showed that it persisted under considerably more generalassumptions. In 1932 Khinchin extended this result to continuous stochastic processes. Thensome foreign scientists (Lévy, Cantelli) refined the solution provided by the law of theiterated logarithm, but its most precise formulation, at least for random continuous processes,was again discovered in Moscow by Petrovsky by means of a remarkable method coveringmany various problems of the modern probability theory, see below.If a magnitude changing under random influences is shown as a point moving along astraight line, on a plane, or in space, we will have a picture of a random motion, or walk of apoint. Many most urgent issues of theoretical physics (problems of diffusion, Brownian