7. Probability and Statistics Soviet Essays - Sheynin, Oscar

14. Kolmogorov studied this particular case in his first work by means of new methods. Ithink that another distinction strongly stressed by Bernstein is less essential. For the Moscowschool, both the discrete pattern of gradually increasing sums of separate terms, and the limitscheme of a random variable continuously changing with a continuous change of a parameter(of time), are stochastic patterns of full value. According to Bernstein’s concept, stochasticterminology is only used for pre-limit discrete schemes. He proved that, as the size ofseparate terms diminishes, the laws of distribution of their sums tend to laws subordinated tothe Fokker – Planck equations, but he did not connect the idea of a continuous series ofrandom variables depending on a parameter (of a random function) with these limit laws.It is true that the limit patterns introduced by the Moscow school possess some paradoxicalproperties (infinite velocities and non-differentiability of the random functions considered),but it should be borne in mind that1) These properties, paradoxical in their limit expression, represent, although sketchily, thequite real characteristics of many physical processes. Take for example particles up to those,sometimes extremely small sizes for which their inertia is felt {begins to be felt}, undergoingBrownian motion considered as a function of time. Their coordinates quite really behave asnon-differentiable functions obeying the Lipschitz condition of the order < 1/2, but notsatisfying it for = 1/2 in any point.It can also be noted on this special occasion that the Brownian motion with the allowancefor the forces of inertia can be studied, as long ago indicated by Kolmogorov, by means ofmore complicated patterns with degenerate Fokker – Planck equations. More essential thanthis last remark, however, is that2) The Moscow school introduced continuous patterns of random processes, characterizedby their compliance with physical reality only under the restriction of an adequate scale. This,however, is a general property of all the mathematical continuous schemes of naturalphenomena. Once their right to exist is denied, it would be natural to declare war also on allthe methods of continuum mechanics which admit that density or the components of velocityare continuous differentiable functions of the coordinates. Indeed, these assumptions becomesenseless on the atomic scale. Finally,3) From the point of view of substantiating the limit theorems of the classical type ondiscrete sums of random variables, their connection with new patterns of random variablescontinuously depending on a parameter does not introduce any arbitrary assumptions of avague nature into the proof of the theorems. After an axiomatic construction of themathematical theory of probability is successfully accomplished, no stochastic considerationsdiffer with regard to rigor in any way from deliberations in pure mathematics: from thelogical point of view, all the relevant stochastic terms are nothing but names of quite definiteand purely mathematical objects. It is therefore not at all more pernicious to turn to thetheory of continuous random processes when proving classical limit theorems, than, forexample, to apply the theory of characteristic functions or parabolic partial differentialequations.15. Interesting work is also being done in Sweden, from where it became recently possibleto obtain a few latest publications of the abovementioned Wold.16. {Kolmogorov did not mention the so-called Continental direction of statistics (Lexis,Bohlmann, Bortkiewicz, Chuprov, Markov).}17. {This statement contradicts Bernstein’s qualification remark (1941, p. 386) to theeffect that he does not reject Fisher’s work altogether.}References1. Bernstein, S.N. (1941), On the Fisherian “confidence” probabilities. (Coll. Works), vol. 4. N.p., 1964, pp. 386 – 393. Translated in Probability andStatistics. Russian Papers of the Soviet Period. Berlin, 2005.