7. Probability and Statistics Soviet Essays - Sheynin, Oscar

are yet no limit theorems corresponding to more general step-wise or mixed processes withcontinuous time governed by integro-differential equations indicated above.6. The Logical Principles of the Theory of ProbabilityFrom the viewpoint of justifying the theory it is natural to separate it into two parts. The first,elementary component is only concerned with patterns involving a finite number of events. The nonelementarypart begins when some random variable is assumed to be able to take an infinite numberof values (for example, when it is being strictly subject to the Gauss law) and extends up to the mostmodern constructions with distributions of probabilities in various functional spaces. It is evident inadvance that the entire second part connected with mathematical infinity cannot claim to have asimpler relation to reality than the mathematics of the infinite in general, i.e., than, for example, thetheory of irrational numbers or the differential calculus.6.1. The Logical Principles of the Elementary Theory of Probability. The events of any finitesystem can be compiled from a finite number of pairwise incompatible events. The matter is thereforecompletely reduced to1) Ascertaining the real sense of the following pattern: Under some conditions S one and only oneof the events A 1 , A 2 , …, A n necessarily occurs with each of them having, given these conditions,probability P(A i );2) Justifying that the pattern in Item 1) always leads toP(A i ) ≥ 0, P(A 1 ) + P(A 2 ) + … + P(A n ) = 1.This problem is known to be solved in different ways. Some believe that the determination of theprobabilities P(A i ) is only scientifically sensible if the conditions S can be repeated indefinitely manytimes and they substantiate, by some method, the concept of probability by the frequency of theoccurrence of the event. Other authors, however, consider that the concept of equiprobability (which,generally speaking, can be introduced without demanding an indefinite repetition of the conditions S)is primary and assume it as the base for defining the numerical value of probability. Both theseapproaches can be subjectively and idealistically colored; apparently, however, they can also beworked out from the viewpoint of objective materialism. The pertinent Soviet literature is scarce; inaddition to introductions in textbooks and popular literature, we only indicate Khinchin’s criticalpaper [13].6.2. The Substantiation of the Non-Elementary Chapters of the Theory ofProbability. When retracting the finiteness of the system F = {A} of events to which definiteprobability {probabilities} P(A) is {are} assigned, it is natural to demand that these events constitutea Boolean algebra with the probability being a non-negative function of its element equal to 1 for aunit of this algebra (Glivenko [6]). It is also possible to demand that P(A) = 0 only for the zero of thealgebra (that is, for the one and only impossible event).The question about countable additivity of the probability does not arise here since for a Booleanalgebra a sum of a countable number of elements has no sense. It is natural, however, to define thesum A k , 1 ≤ k < ¦ , as an event A for whichlim P[( A k – A) (A – A k )] = 0 as n § ¦ .Then countable additivity will certainly take place.The Boolean algebra of events can be incomplete in the sense that a countable sum of its elementsdoes not always exist. Then, however, it can be replenished and this operation is uniquely defined. Itis as natural as the introduction of irrational numbers. On principle, Glivenko’s concept just describedis the most natural. However,1) The axiomatics of the theory of probability understood as the theory of complete normedBoolean algebras is rather complicated.2) In this theory, the definition of the concept of random variable is too complicated.