1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

i. e., if many observational series can be obtained under similarstatistically homogeneous conditions. More often, however, we haveonly one such series, distributions of probabilities certainly can not bereconstructed and the model of an n-dimensional distribution isabsolutely useless. However, if we assume that the joint distribution ofthe magnitude ξ 1 , ξ 2 , is the same as that of ξ 2 , ξ 3 , of ξ 3 , ξ 4 , etc, then thepairs (ξ 1 , ξ 2 ), (ξ 2 , ξ 3 ),..., (ξ n−1 , ξ n ) provide many realizations, althoughperhaps not mutually independent, of that bivariate distribution. Such adistribution is therefore determinable in principle.It is convenient to generalize somewhat the mathematical model. Letus consider a sequence of random variables infinite in both directions... ξ −1 , ξ 0 , ξ 1 , ξ 2 , ..., ξ n , ξ n+1 , ... (1.10)called a stochastic process. We assume that theoretically there existdistributions of probabilities of any finite set{ξ α , ξ β , ξ γ } (1.11)of random variables. Our observational series (1.9) is a part of theinfinite sequence (1.10) and only allows us to reach some conclusionsabout that whole process if the model includes a rule representingdistributions of magnitudes (1.11) with negative and large positivesubscripts through the distribution of the observed variables (1.9).Without such a rule the model of a stochastic process is useless.In the most simple and most natural case the condition ofstationarity is imposed: for any τ the distribution of the variables (ξ α+τ ,..., ξ γ+ τ ) coincides with that for τ = 0. The model of a stochasticprocess consists in that [now] we consider our observations (1.9) as apart of the realization (1.10) of a stationary stochastic process.When assuming a model of a stochastic process, only bivariatedistributions are usually applied and in addition only the correlationbetween the different values of that process are studied. It ought to besaid that in spite of the popularity of the concept of stochastic process,only quite a few examples can be cited in which it allowed to describeadequately the statistical properties of observational series. Mostpublications begin by stating that a pertinent stochastic processspecified in such and such a way is given, but there really are only afew works where these specifications are indeed determinedtheoretically or experimentally.The theory of stochastic processes is here suitable for solvingabstract problems: what will happen if a white noise of a givenintensity influences some system. Such problems, however, onlyindirectly bear on the real behaviour of a system because under realconditions it is not likely the white noise that influences the system, –it does not even concern a stochastic process (lack of statisticalhomogeneity). But meanwhile often no one studies what is reallyacting on the system because such investigations are complicated,difficult and expensive so that it is much easier to restrict the attentionto arbitrary prior assumptions.54