7. Probability and Statistics Soviet Essays - Sheynin, Oscar

terms are connected by a recurrent linear relation which takes place with a high probability.These terms are therefore situated in the vicinity of certain sinusoids (or of combinations ofsuch curves) which constitutes the limiting sinusoidal law and very interesting models ofsuch series were constructed (Slutsky, Romanovsky). Khinchin recently proved that eachstationary series obeys the law of large numbers and this fact certainly considerablystrengthened the interest in them. Gelfond and Khinchin, in yet unpublished contributions,studied the properties of the Gram determinants for stationary series.8. The interest in the so-called congestion problems, that is, in stochastic investigationsconnected with the running of generally used plants, essentially increased mostly inconnection with the development of automatic telephony. By now, these studies resulted inthe creation of a special theoretical chapter of the doctrine of probability, and we aretherefore mentioning them here. The Moscow school (Kolmogorov, Khinchin) published anumber of pertinent writings which theoretically solved sufficiently general problems.[9] Finally, we ought to say a few words about some isolated works. In spite of theirapplied nature, it is difficult to pass over in silence Bernstein’s remarkable investigations ofheredity possessing considerable theoretical interest {contradiction!}. Kolmogorov recentlysolved a number of separate, and, again, theoretically important related problems. He alsostudied the general forms of mean values satisfying definite natural demands. His workoccasioned essential response from foreign scientific circles.We are now concluding our essay, and we repeat that it is very incomplete. We did notgive their due to all the works mentioned, but we still hope that we have attained our mainaim by showing that Soviet mathematics, in spite of the tenfold efforts exerted by ourEuropean comrades in competition, is firmly holding that banner of championship inprobability theory which the pre-revolutionary Russian science had already deserved.Notes1. {The civil war ended in 1920 and scarcely any serious work had begun until then, oreven until several years later.}2. {The official term was Great October [new style: November 7] Socialist Revolution.Contrary to Russian grammatical rules, all four words were capitalized.}3. A.N. Kolmogorov. On Some Modern Currents in the Theory of Probability 1934. (Proc. Second All-UnionMathematical Conference 1934), vol. 1. Leningrad – Moscow, 1935, pp. 349 – 358[Introduction] The first, classical period in the development of the theory of probabilityessentially ended with the investigations of Laplace and Poisson. Then, the theory wasmostly engaged in the calculation of the probabilities of various combinations of a finitenumber of random events. Entirely in accord with the problems studied, its mathematicaltools were mainly combinatorial analysis, difference equations, and, when solving these, themethod of generating functions.Owing to their fundamental research, Chebyshev, Markov and Liapunov initiated a newdirection. During that {new} period the concept of random variable occupied the centralposition. New analytic machinery for studying these variables, substantially based on thenotion of expectations, on the theory of moments and distribution functions was created. Themain objects of examination were sums of an increasing (but always finite) number ofrandom variables, at first independent, and later dependent. Mises (1919) developed acomplete theory of n-dimensional distribution functions for n random variables depending