7. Probability and Statistics Soviet Essays - Sheynin, Oscar

7. Grenander, U., Rosenblatt, M. (1955), Statistical Analysis of Stationary Time Series.New York.14. A.N. Kolmogorov. Issues in the Theory of Probability and Mathematical StatisticsReport Made at the Branch of Mathematics, Academy of Sciences of the Soviet UnionVestnik Akademii Nauk SSSR, No. 5, vol. 35, 1965, pp. 94 – 96 …[1] I would like to begin the review of the present state and the main directions of thedevelopment of probability theory and mathematical statistics by mentioning that vol. 4 ofS.N. Bernstein’s (Coll. Works. N.p., 1964) containing {almost} all hiswritings on the theory of probability and statistics has appeared. Something is therenowadays interesting mainly for the history of science since it had been included, in a moreclear form, in textbooks, but the store of ideas, far from being exhausted, and sometimesinsufficiently known to young researchers, is also vast.Issues belonging to the domain of limit theorems going back to Chebyshev and Liapunov,and essentially developed by Markov and Bernstein in the direction of studying dependentvariables, which seemed for some time to be exhausted, experiences a period of newflourishing. V.A. Statuljavicus’ report “Limit theorems in boundary value problems andsome of their applications”, read out on 29 Oct. 1964 at the General meeting of theAcademy’s section of mathematics on the theory of probability and mathematical statistics,was devoted to some pertinent issues.[2] At the same meeting, A.A. Borovkov reported on a cycle of works which goes back toanother current in the field of limit theorems apparently originated by Cramér, – to the socalledtheorems on large deviations. Borovkov’s works are beginning to show importance forapplications in mathematical statistics and it is worthwhile to dwell somewhat on this point.The simplest typical problems of mathematical statistics contain two parameters, significancelevel (the admissible probability of a mistaken judgement) and n, the number ofobservations. The approach based on limit theorems of the Chebyshev type corresponds to alimiting transition (n ) with a constant . However, in practice n is often of the order ofonly a few hundred, or even a few dozens, and the significance level is usually chosen in theinterval from 0.05 to 0.001. The number of problems demanding the guarantee of a very highreliability, i.e., of a very small , will probably increase ever more. Therefore, the formulasof the theory of large deviations corresponding to the asymptotic case at 0 are more oftenapplicable.[3] Markov originated the study of a vast class of stochastic processes now everywherecalled Markov processes. After his time, our country continued to play a very large part indeveloping this direction, especially owing to the Dynkin school. Problems of obtaining thewidest possible general conditions for the applicability of the main theorems of the theory ofMarkov processes; of ridding the theory of superfluous assumptions are still unsolved.However, I believe that the most essential work is here the search for new issues even if thesedo not demand the use of excessively refined mathematical tools but cover a wider field ofapplications.In particular, an urgent issue is the study of only partly observable Markov processes, i.e.,processes of the type of x(t) = {x 1 (t); x 2 (t)} where only the first component, x 1 (t), isobservable. R.L. Stratonovich, in his theory of conditional Markov processes, formulatedextremely interesting ideas about the approaches to solving the problems here encountered.Regrettably, his works sometimes lack not only any special mathematical refinement, but areoften carried out on a level that does not guarantee a reasonable rigor not absolute, but