7. Probability and Statistics Soviet Essays - Sheynin, Oscar

a precise interpolation of a stationary sequence of vectors. The corresponding extrapolationalproblem is still not quite effectively solved.Pinsker & Yaglom [1] constructed a systematic theory of processes with n-th stationaryincrements (also see Yaglom [17; 18]). In essence, this theory ought to fit in with the theoryof stationary generalized functions as created according to Ito and Gelfand (§1) as aparticular case. Pinsker, whose work [3] I have already mentioned in the Introduction,provided a simple spectral representation of the amount of information per unit time inprocess (t) with respect to process (t) for Gaussian processes. Kolmogorov, in his essay[160], explained how to apply this formula to calculate the velocity of creating messages fora given precision of transmission and the channel capacity. Pinsker [5] derived one-sidedestimates of the same magnitudes also for non-Gaussian processes.Obukhov [7; 8] and Yaglom [28], in connection with the development of the theory ofhomogeneous and locally homogeneous (in particular, of isotropic and locally isotropic)turbulence, worked out the spectral theory of homogeneous vector fields havinghomogeneous increments. It can have many other applications as well. Chiang Tse-pei(1957) initiated the study of the corresponding extrapolation problems (from semi-space tothe entire space).The main difficulty in the engineering applications of the theory of stationary processes(see the books of Bunimovich [1] and Levin [7]) is encountered when considering non-linearproblems. If the sought process is connected with the given processes by linear differential orintegral equations, its spectrum is calculated by issuing from the spectra of these latter. This,however, is not so if the connections are non-linear, and even when calculating the spectrumof a given process we have to apply more subtle characteristics of the given and theintermediate processes. Theoretically, we can use the characteristic functional, and, in mostcases, moments of the higher order. However, in spite of numerous works published in thissphere, any harmonious general non-linear theory of stationary processes is still lacking. Inapplications, a prominent part is played by considering a real stationary processλ 1 (t) = e i t Φ ( dλ)together with an adjoint process 2 (t) which leads to the concepts of envelope v(t) =22ξ1( t)+ ξ2( t)and phase. Approximate methods for calculating the spectrum of theenvelope are worked out.In applications, much attention is given to the calculation of such magnitudes as the meannumber of ejections (of (t) passing beyond pre-assigned limits) and the distribution of theirdurations. The first problem is easily and definitively solved, but the methods offered forsolving the second one are still very imperfect (see Bunimovich [1] and Kuznetsov,Stratonovich & Tikhonov [18]).3. Markov Processes with Continuous TimeThe transition probabilitiesP s t (x; M) = P[(t) ∈M|(s) = x], s ≤ tdescribing the transition from state x into the set of states M during the interval of time from sto t generate a system of operators F t = H s t F s which transform the distribution ofprobabilities F s at time s into distribution F t at time t. These operators are connected by therelation H s t = H u t H s u , s ≤ u ≤ t.