7. Probability and Statistics Soviet Essays - Sheynin, Oscar

Slutsky; Obukhov; and others have been developing the appropriate general mathematicalproblems. The main issue here is to avoid fictitious dependences artificially introduced in ausually very restricted material (30 – 100 observations) 12 . This method can undoubtedly beuseful when being adequately careful, but even from a purely mathematical side far from allthe appropriate circumstances encountered when applying it are sufficiently ascertained.The statistical determination of distribution functions consists in the following. Adistribution function F(x) is unknown; n independent observations corresponding to it aremade and an empirical distribution step-function F n (x) is computed. It is required to ascertain,to what extent may we form an opinion about the type of the function F(x) by issuing fromF n (x). Glivenko subordinated the very problem about F n (x) tending to F(x) to a general law oflarge numbers in functional spaces which he {also} established. For statistics, it was,however, important to have as precise estimates of the deviations of F n (x) from F(x) aspossible. Kolmogorov had provided the first asymptotic formulas for the law of distributionof these deviations 13 whereas Smirnov deeply and thoroughly studied all the relationsbetween these functions by far exceeding all the previous results. His findings have mostvarious applications in statistical investigations.We have indicated two directions of research where the results of Soviet mathematiciansoffered a considerable contribution to the general development of mathematical statistics. Inconnection with the permanently occurring practical problems very many more specialinvestigations were also made. Among these we point out for example the works of A.M.Zhuravsky on the statistical determination of the composition of minerals; and B.V.Yastremsky’s studies of the application of sampling. It ought to be said, however, that theassistance rendered by mathematicians to the applications of mathematical statistics and thetheory of probability was until now of a somewhat casual and amateurish nature and wasmostly directed either by personal links or special interests of the individual researchers inapplied issues connected with the essence of their general theoretical work. The furtherdevelopment of applied studies will undoubtedly require the creation of adequatecomputational tools, the compilation and publication of tables, and the design of {special}devices 14 .The creation of a scientific institution which would be able to shoulder all theseduties and to attend systematically to the requirements of an applied nature is a problem forthe near future.Notes1. {Gauss should not have been mentioned here.}2. {Apparently, 1919.}3. Beyond this special sphere these methods subsequently proved themselves essential fora rigorous formal justification of the theory of probability including extensions that havebeen required by its further development.4. Feller has recently precisely formulated the necessary and sufficient conditionscorresponding to this idea somewhat vaguely expressed by me.5. We are usually interested in low probabilities of the order of 1/1,000 or 1/10,000. Forreliably estimating them allowing for the existing expressions of the remainder terms thenumber of observations ought to be essentially larger than 10 6 or, respectively, 10 8 .6. Among these, in addition to the abovementioned Lévy, Cramér and Feller, is Mises whoachieved fundamental results. {Kolmogorov did not mention Cramér.}7. He generalized Birkhoff’s ergodic theorem for dynamic systems.8. In accordance with its intention the spectral analysis of stationary stochastic processesadjoins Wiener’s generalized harmonic analysis in the theory of functions.9. Keller’s research was going on independently of Khinchin’s and Slutsky’s work as wellas of that of Wiener (above).10. {Kolmogorov apparently referred to his earlier essay also translated in this book.}