7. Probability and Statistics Soviet Essays - Sheynin, Oscar

a) The theory of stochastic processes (Finetti, Hostinsky, Hadamard, Mises, Kolmogorov,Fréchet, Khinchin, Lévy).b) The theory of random functions (Wiener, Slutsky, Lévy).3) New analytic toolsa) Equations of stochastic processes (Kolmogorov, Hostinsky, Fréchet, Bernstein,Pontriagin).b) Characteristic functions and moments in infinite-dimensional and functional spaces(Khinchin).c) New methods of proving limit theorems (Kolmogorov, Petrovsky, Bernstein, Khinchin,Bavli).1a. As stated above, all the problems of probability theory considered up to the lastdecades can be reduced to the study of a finite number of random variables. All theprobabilities encountered here will be determined if we provide an n-dimensional distributionfunction of these variables. In particular, Mises, in his course in the theory of probability(1931), systematically kept to the mentioned restriction. In his terminology, this wasexpressed by assuming that the set of indications (Merkmalmenge) of an arbitrary collectivewas always a set of points of an n-dimensional space.It is the most natural to perceive a distribution function in an n-dimensional space as anadditive function in a domain of that space. The mathematical interpretation of a problem inthe theory of probability of the indicated classical type depends exclusively on thecorresponding distribution function. Therefore, if the only aim of the axiomatic theory ofprobability is the most compact and clear enumeration of the logical assumptions ofsubsequent mathematical constructions, then, when dealing with problems of the classicaltype, it would be simplest to select directly as the axioms the characteristic properties ofdistribution functions (non-negativity; additivity; and their being equal to 1 for the completespace). According to my deep conviction, the axiomatics of probability theory cannot haveany other goals because the question about the applicability of a given mathematical patternto some concrete physical phenomena cannot in essence be solved by the axiomatic method.Mises’ attempt clearly illustrates this idea. To confine his construction within the boundariesof fixed axioms, he is compelled only to postulate the approach of the frequencies to certainlimits as the trials are unboundedly continued, without saying anything about when,beginning with what finite number of repetitions, may we conclude that the formerpractically coincide with the latter. Indeed, an answer to this question can only be providedafter going beyond the boundaries of a rigorously formal mathematical thinking. Thus, theMises axioms, irrespective of the connected intrinsic difficulties, are, one the one hand, notneeded for justifying the mathematical theory, and insufficient for substantiating itsapplicability on the other hand.My system of axioms [1] is a direct generalization of the properties of distributionfunctions listed above. This generalization allows us to cover all those new non-classicalproblems described in the Introduction. I only indicate here one point concerning theaxiomatic construction of the main notions of the theory of probability, – that point, which,as it seems to me, still requires to be developed.In the applications, researchers often consider conditional probabilities which aredetermined under the restriction that some random variable x took a definite particular valuex = a. If x has a continuous law of distribution, the elementary method of determining theconditional probability P x=a (A) of event A given that x = a,P x=a (A) = P(A|x = a)/P(x = a),