7. Probability and Statistics Soviet Essays - Sheynin, Oscar

Such new demands were formulated in the second half of the 19 th century, first andforemost owing to the development of statistical methods in the social and biologicalsciences. The complex of theories that is usually understood and taught as mathematicalstatistics took shape on these very grounds. The theory of probability enters the sphere ofissues mainly when determining whether a {given} restricted number of observations issufficient for some deductions. Exactly here we may perceive a certain (naturally relative)specificity of the stochastic problems of mathematical statistics. In the 20 th century, thesupremacy of the British (Karl Pearson, Fisher) and partly American schools in mathematicalstatistics had definitely shown itself. An increasing number of tests for the compatibility of agiven finite series of observations with some statistical hypothesis, which were offered onvarious special occasions, has led during the last years to an intensive search for generalunifying principles of mathematical statistics considered from this very point of view ofhypotheses testing (for example, in the works of Neyman and E.S. Pearson). Here, Sovietmathematicians accomplished a number of remarkable special investigations (§3), but, for us,the development and reappraisal of the general concepts of mathematical statistics, which arenow going on abroad under a great influence of the idealistic philosophy, remains a matterfor the future.In statistical physics, the issue of the sufficiency of a restricted number of observationswithdraws to the background. Here, the directly observed quantities are most often the resultsof a superposition of a great number of random phenomena (for example, on a molecularscale). The material for testing hypotheses on the course of separate random phenomena isthus collected independently of us in quite a sufficient measure. On the other hand, statisticalphysics makes especially great demands for developing new patterns, wider than the classicalones, of the course of random phenomena. The first decades of the 20 th century had beenespecially peculiar in that the physicists themselves, not being satisfied with the possibilitiesprovided by the classical theory of probability, began to create, on isolated particularoccasions, new stochastic patterns. A number of biologists and technicians encountered thesame necessity. For example, studies of the theory of diffusion had led Fokker and Planck tothe construction of the apparatus of differential equations which were later discoveredalready as general differential equations of arbitrary continuous stochastic processes withoutaftereffect (§2). Fisher introduced the same equations quite independently when studyingsome biological issues. To such investigations, that anticipate the subsequent development ofthe general concepts of probability theory, belong also the works of Einstein andSmoluchowski on the theory of the Brownian motion, a number of studies accomplished byAmerican technicians (T. Fry et al) on the issues of queuing connected in particular withproblems of exploiting telephone networks, etc. Finally, the theory of probabilityencountered most serious problems in the direction of stochastically justifying the so-calledergodic hypothesis that underpins thermodynamics. Only around 1930 mathematiciansearnestly undertook to systematize all this material. It is the most expedient to unite all thethus originated investigations under the name of general theory of stochastic processes.After ascertaining the situation in which probability theory developed during the lasttwenty years, we go on to review, under three main heads, the pertinent accomplishments ofSoviet scientists.1) Extension of the classical investigation of limit theorems for sums of independent andweakly dependent random variables.2) General theory of stochastic processes.3) Issues in mathematical statistics.Irrespective of this methodical division, we believe that considerable work on the theory ofprobability in the Soviet Union began roughly in 1924 – 1925. S.N. Bernstein’s fundamentalresearch (see below) appeared in 1925 – 1926, and it alone would have been sufficient forconsidering that the traditions of Chebyshev, Markov and Liapunov were worthily continued.