7. Probability and Statistics Soviet Essays - Sheynin, Oscar

securing against mistakes (A.N. Krylov’s expression) 1 . A.D. Ventsel described in his reporthow a certain part of the theory of conditional Markov processes can be constructed with duerigor.[4] The spectral theory of stationary stochastic processes whose rigorous foundation waslaid in our country by {the late} Khinchin, is being intensively developed. Here, specialattention, perhaps under the influence of Wiener’s ideas, is paid now to the attempts atcreating a spectral non-linear theory. This is indeed essential since the specialists in the fieldsof radio engineering, transmission of information, etc are inclined to apply spectral notionswhereas only the linear theory, absolutely inadequate for many practically importantapplications, is yet mathematically worked out for the continuous spectra typical for thestochastic processes.[5] In the area of information theory our scientists had to catch up with science abroad. Wemay assume that now this delay is made up for, and the works of Khinchin and of therepresentative of our younger generation, R.L. Dobrushin, have already occupied aprominent place in international science.By its nature, information is not an exclusively stochastic notion. The initial idea ofinformation as the number of binary symbols needed for isolating a certain object fromamong a finite number of objects has nothing in common with the theory of probability.Stochastic methods now only dominate the higher sections of the theory of information. It ispossible, however, that the relationship between the two theories will radically change. I donot want to dwell here on this viewpoint (I am personally ever more attracted to it) accordingto which these relations may be reversed as compared with the present situation so that notprobability theory will serve as a basis of the higher sections of the theory of information, butthe concepts of the latter will form the foundation of the former.[6] I only note the origin of the new branch of the theory of dynamic systems, i.e., of thegeneral theory of non-stochastic rigorously determinated processes where the ideas of thetheory of information (beginning with the informational idea of entropy) play the main part.Extensive analogies between dynamic systems possessing the property of intermixing withstochastic processes were understood long ago. Now, however, in the works which I hadbegun and which V.A. Rokhlin and especially Ya.G. Sinai have continued, these similaritieswere essentially deepened. In particular, Sinai proved, under broad assumptions, and forsome quite classical models (elastic balls in a box), the long-standing hypothesis on theasymptotically normal distribution of the sojourn periods for different sections of the phasespace. For classical dynamic systems, defined by vector fields on compact manifolds, the twoextreme instances, the almost-periodic case being studied by me and V.I. Arnold, and thecase of K-systems with intermixing, are apparently the main ones in some sense.[7] In mathematical statistics, in spite of many splendid investigations accomplished in theschools of N.V. Smirnov and Yu.V. Linnik, the work of Soviet mathematicians is yet farfrom being sufficient. As it seems, this situation is caused by the fact that the development ofmathematical statistics is closely connected with the experience of direct contact with actualstatistical material, whereas, for qualified Soviet mathematicians, such work with real datastill remains although not rare, yet incidental and somewhat casual. Linnik reported on hisremarkable accomplishments in solving difficult analytical problems appearing inmathematical statistics. Work on publishing mathematical tables required in statisticalpractice and on compiling a number of new tables is going on on a vast scale at the SteklovMathematical Institute under Smirnov and L.N. Bolshev 2 .