7. Probability and Statistics Soviet Essays - Sheynin, Oscar

Markov chains correspond to the case in which the states of a system are only consideredat integer values of t and the number of the possible states is finite. Here, all is reduced to theprobabilities P ik (t) of the transition from state i to state k during the interval of time betweenthe moments t and (t + 1). The more general probabilities P ik (s; t) of the same transitionbetween moments s and t > s are expressed through P ik (t). At present, a large number ofgeneralized Markov chains have been studied, and I discuss some of them in §3a. Thefollowing pattern covers all of them.Let E be the set of the possible states of the system, F, – the distribution function ofprobabilities in E. Then, for each interval of time from moment s to moment t > s there existsan operator H st (F) = F 1 by whose means we can determine the distribution function F 1 atmoment t given F at moment s. This operator is inevitably linear and unitary. In addition, itobeys the equationH su = H tu H st (1)for any s < t < u. Equation (1) is indeed the main equation of stochastic processes withoutaftereffect. For the case of Markov chains it takes the well-known formP ik (s; u) =jP jk (t; u)P ij (s; t).The general solution of equation (1) seems to be very difficult even for separate particularcases. The most important instance is the time-homogeneous case in which H st = H t–s and inwhich, accordingly, equation (1) becomesH s+t = H s H t . (1)Equation (1) shows that the matter indeed concerns the determination of the general formof a one-parameter group of unitary operations H t in the space of distribution functions F.The natural assumption that H t = e tU always leads to wide and important cases of solutions ofthe equation (1). It remains unknown whether a convenient method of forming a generalsolution, when the symbol U is understood in a sufficiently general sense, is possible.In addition to stochastic processes without aftereffect, another class of such processes withaftereffect but stationary (where all the distributions persist under the change from t to t = t+ a) was also deeply studied. Khinchin proved a profound theorem for stationary processesgeneralizing the Birkhoff ergodic proposition. Hardly anything is known about nonstationaryprocesses with aftereffect.Introductory literature1. Hostinsky, B. Méthodes générales du calcul des probabilités. Mém. Sci. Math., t. 52,1931, pp. 1 – 66.2. Kolmogoroff, A. Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung.Math. Ann., Bd. 104, 1931, pp. 415 – 458.Further bibliographic indications are in3. Khintchine, A. Asymptotische Gesetze der Wahrscheinlichkeitsrechnung. Berlin, 1933.2b. A systematic study of random functions is just beginning. It is certain, however, that adeeper theory of stochastic processes will be essentially based on the notions connected withrandom functions. Owing to the incompleteness of the axiomatics of probability theory,many investigations (Wiener, Slutsky) were restricted to considering the values of functionsat a finite number of points. In this case, even the formulation of the problem concerning theconditions of continuity, integrability, differentiability of a function, etc was impossible.