7. Probability and Statistics Soviet Essays - Sheynin, Oscar

day portrait of this discipline has no features, nor has its workshop a single range of problemswhose origin was not initiated by Soviet mathematics. And the level of our science is bestdescribed by noting that European mathematicians are until now discovering laws known to,and published by us already five – ten years ago; and that they (sometimes includingscientists of the very first rank) publish results which were long ago introduced into ourlectures and seminars and which we have never made public only because of consideringthem shallow and immature if not altogether trivial.It is interesting to note that after the October upheaval 2 our probability theory underwent aconsiderable geographical shift. In former times, it was chiefly cultivated in Leningrad {inPetersburg/Petrograd} but during these 15 years Leningrad {Petrograd until 1924} did notoffer any considerable achievements. The main school was created in Moscow (Khinchin,Kolmogorov, Glivenko, Smirnov, Slutsky) and very substantial findings were made inKharkov (Bernstein) and Tashkent (Romanovsky).Before proceeding to our brief essay on those lines of development of the theory ofprobability which we consider most important, we ought to warn our readers that it is not ourgoal to provide even an incomplete list of Soviet accomplishments in this sphere; we aim attracing the main issues which occupied our scientists and at indicating the essence which wecontributed to the international development of our science, but we do not claim anycomprehensiveness here. Then, we ought to add a reservation to the effect that we restrict ourdescription to those lines of investigation which have purely theoretical importance, and weleave completely aside all research in the field of practical statistics and other applications ofprobability in which Soviet science may also take pride. We consider the following issues ashaving been basic for the development of the Soviet theory of probability.1. Investigations connected with the so-called limit theorem of the theory of probability,i.e., with the justification of the Gauss law as the limiting distribution for normed sums ofrandom variables. Formerly, almost exclusively considered were the one-dimensional caseand series of mutually independent random variables. Nowadays we are able to extend thelimit theorem, on the one hand, to the many-dimensional case, and, on the other hand, toseries of mutually dependent variables. Soviet mathematicians and Bernstein in the first placeboth initiated these generalizations and achieved the most important discoveries {here}.Bernstein accomplished fundamental results in both directions; Kolmogorov arrived at someextensions; Khinchin justified normal correlation by the direct Lindeberg method. Thissphere of issues met with a most lively response in the European literature and continues tobe developed both at home and abroad.2. For a long time now, the theory of functions of a real variable, that was worked out atthe beginning of our {of the 20 th } century, compelled mathematicians to feel a number ofdeep similarities connecting the stochastic concepts and methods with the main notions of themetric theory of sets and functions. The appropriate process of modernizing the methodologyof probability theory, its notation and terminology, is one of the most important aspects ofthe modern issues of our scientific domain. This process is far from being completed, but wealready feel with complete definiteness how much it is offering to probability theory, – howmuch its development fosters the unity and the clarity of the scientific method, the harmonyand the visibility of the scientific building itself. With respect to their level, even the mostpowerful stochastic schools (for example, the Italian school) insufficiently mastering themethods of the function theory become noticeably lower than those (the French, the Moscowschool) where these methods are entrenched. It is therefore quite natural that Moscowmathematicians, most of whom had developed in the Luzin school, have marched in the frontline of this movement. Slutsky should be named here in the first instance, then Glivenko andKolmogorov. In particular, the last-mentioned had revealed, with exhausting depth and