7. Probability and Statistics Soviet Essays - Sheynin, Oscar

Kuznetsov, Stratonovich & Tikhonov [18; 20; 21] called the factors a n and b n momentousand correlational functions respectively. They began to realize a wide program of applyingthis tool to solving concrete problems (also see Cherenkov [1]).A more elementary theory only using the first moments A 1 (f) and B 1 (f) and the secondcentral moments B 2 (f) is widely applied in contributions to technical sciences. The quadraticfunctional B 2 (f) is reduced to a sum of squares by an adequate choice of the coordinates; thisis simply Pugachev’s canonical expansion [13]. Pugachev’s book [21] summarizes theeffective methods and the experience in applying the theory of random functions inengineering 1 .The theory of distributions in infinite-dimensional linear spaces, when compared with thetheory of finite-dimensional distributions, suffers from some defects. Thus, not anycontinuous positive-definite functional H(f) having H(0) = 1 is a characteristic functional ofsome distribution. Even for the Hilbert space the additional conditions which should haveensured the existence of a corresponding distribution remain scarcely effective. This factdoes not, however, persist in the theory of generalized random functions also having manydirect applications. A number of findings pertaining to the theory of generalized functionswere contained in Gelfand’s short note [65].2. Stationary Processes and Homogeneous Random FieldsWe (G&K, §§4.3 – 4.4) briefly mentioned the works of Khinchin, Kolmogorov, Zasukhinand Krein on the spectral theory of stationary processes. During that previous period themain achievements were as follows. The contributions of Slutsky on stationary sequencesand Wiener’s general harmonic analysis brought about the understanding that stationarity ofa process automatically leads to the possibility of a spectral representation. Khinchin [72]provided an appropriate harmonious and very simple general spectral theory of stationary (inthe wide sense) processes that initiated a large section of the modern theory of probability.Kolmogorov remarked that from a formal mathematical point of view this theory was a directcorollary of the spectral theory of one-parameter groups of unitary operators. This enabledhim to offer a simple exposition of the findings of Khinchin and Cramér, and to construct,issuing from the works of Wold, a spectral theory of extrapolation and interpolation ofstationary sequences [84; 90; 92].After its continuous analogue was discovered; and after itwas supplemented with the somewhat later Wiener theory of filtration, it acquired essentialimportance for radio engineering and the theory of regulation. Kolmogorov [85; 86] providedthe theory of processes with stationary increments in a geometric form. Zasukhin [1] solvedsome problems of the many-dimensional spectral theory (see a modern explication of hisfindings in Rosanov (1958)). Finally, Krein [85] offered the abovementioned continuousversion of the theory of extrapolation. In particular, he established that the divergence of theintegral+∞log1+− ∞f ( λ)d λ2λwith f() being the spectral density was necessary and sufficient for singularity (i.e., for thepossibility of precise extrapolation). Yaglom’s essay [12] described the further developmentof the spectral theory of stationary processes up to 1952. Krein [134] and Yaglom [11; 15;19] devoted their writings to issues in extrapolation and filtration for stationary onedimensionalprocesses. Rosanov [2; 3] studied the many-dimensional case for sequences 2 ; inparticular, he discovered an effective necessary and sufficient condition for the possibility of