7. Probability and Statistics Soviet Essays - Sheynin, Oscar

functions 1 (t), 2 (t), …, n (t), and stationarity is restricted by demanding that E k (t) = A k do notdepend on time andE[ i (t) j (t + )] = B ij ( )only depend on .Without loss of generality it is possible to assume that A k = 0 and B kk = 1, k = 1, 2, …, n. Then B ij ( )is nothing but the correlation coefficient between i (t) and j (t + ). Khinchin [32] outlined thegeneral patterns emerging from these propositions of the correlation theory of random processes.Kolmogorov [47] reviewed its further development in which Slutsky, he himself, V.N. Zasukhin andM.G. Krein had participated.4.5. Other Types of Random Processes. A number of studies is devoted to various special typesof random processes that do not belong to the patterns described above. These include the so-calledgeneralized chains introduced already by Markov which can be reduced (as the number of statesincreases) to usual simple Markov chains. Stochastic bridges 6 introduced by Bernstein [21], whichgeneralize the notion of Markov process, would have deserved a more detailed investigation. It isnatural that the study of stationary processes leads to processes with stationary increments.Bogoliubov carried out important investigations of the limiting transitions from some types ofrandom processes to other ones (for example, he studied the conditions for a limiting transition of anon-Markov process depending on a parameter into a Markov process).5. A New Approach to Limiting Theorems for Sums of a Large Number ofTermsIt is well known from the history of the classical analysis that a limiting transition from relationsbetween finite differences to the corresponding differential expressions often leads to considerablysimpler results as compared with those accomplished by a direct investigation of the initial relations.In a similar way, most asymptotic formulas derived by long reasoning in the classical studies of sumsof a large number of random terms, or of results of a large number of trials, appear as precisesolutions of naturally and simply formulated problems in the theory of random processes withcontinuous time. This idea that goes back to Poincaré and Bachelier is convincingly developed bymeans of simplest illustrations, see for example Khinchin [33]. It offers a leading principle forconstructing new proofs and for discovering new formulations of the limit theorems of the classicaltype. Some foreign authors from Courant’s school (Pólya, Lüneberg) were first to apply such amethod and Kolmogorov [32] provided its general outline. In his abovementioned works [13; 18] healso applied it for proving the classical Liapunov limit theorem and limit theorems of a new type forsums of independent terms.The method was further developed according to two methodologically different viewpoints.Bernstein systematically develops the method of stochastic differential equations but does not connect{it with} the limiting differential equations of the limiting probability-theoretic pattern of a randomprocess with continuous time. On the contrary, Petrovsky and Khinchin always bear in mind thislimiting random process.As to actual results, Bernstein [22; 27; 31; 41] examined considerably more fully the difficultiesthat occur in cases of a non-compact phase space (such, for example, as even the usual real straightline with which he indeed has to do), whereas Petrovsky [1] and Khinchin [33] more fully studiedproblems with boundaries.We note that the totality of the available theoretically rigorous contributions of this type does not atall cover the cases in which similar limit transitions are made use of in applications. Further intensivework in this direction is therefore desirable.The abovementioned works of the three authors are mostly concerned with limit theoremsconnected with continuous random processes governed by the Fokker – Planck equations. OnlyKhinchin [33] generalizes the Poisson limit theorem for step-wise processes with continuous time.This theorem was the foundation of all the investigations of limit theorems for sums leading toinfinitely divisible laws and expounded in detail in §1. It should be noted in this connection that there