7. Probability and Statistics Soviet Essays - Sheynin, Oscar

nB n =wherei = 1nj = 1( n)( n)ibjb r ij (n) (2.1.2)c ij (n) = E[( i (n) – E i (n) ) ( j (n) – E j (n) )]are the mixed moments of the second order, andr ij (n) = [c ij (n) /( n)( n)ibjb ] = R( i (n) ; j (n) )are the correlation coefficients between i (n) and j (n) .Formula (2.1.2) enables to elicit from the classical conditionlim B n = 0 as n ¢ £ (2.1.3)a number of other sufficient conditions for the normal stability of the sums n expressed through thevariances of the terms b i (n) and the coefficients r ij (n) . Bernstein and Khinchin engaged in this subject;thus, Bernstein [41] provided the following sufficient condition for the arithmetic means of thesequence of variables i :var i ≤ C, R( i ; j ) ≤ (|j – i|). (2.1.4)Here, C is a constant, and the function (m) tends to zero as m ¢ £ . It is natural, however, that, as wealready noted in the case of independent terms, the problem cannot be solved in a definitive way onlyby considering the second moments.In a few of his works Khinchin studied the applicability of the strong law of large numbers tosequences of dependent variables. Here, however, even the question of what can the second momentsensure is not ascertained in full 3 . Fuller findings are obtained for the cases in which the terms appearas a result of a random process of some special (Markov or stationary) type, see below.2.2. The Classical Limit Theorem. Bernstein [11; 13; 18; 40] continued Markov’s research of theconditions for the normal attraction to the Gauss law of sums of an increasing number of weaklydependent terms. His results relate to the pattern of Markov chains, see below. The formulations ofhis subtle findings directly expressed by demanding a sufficiently rapid weakening of the dependenceof the terms when the difference between their serial numbers increases are rather complicated. Weshall only note that for this problem the estimation of the dependence by the correlation coefficients istoo crude. Instead of applying them we have to demand either complete independence for terms withsufficiently large serial numbers or to require that the conditional moments of the first and the secondorder of the terms, when the values of the previous terms are fixed, should little differ from theunconditional moments. The meaning of the conditions of the second kind became quite clear in thecontext of the theory of stochastic differential equations that emerged later.3. The Ideas of the Metric Theory of Functions in Probability TheoryAs a science devoted to a quantitative study of the specific domain of the random, the theory ofprobability is not a part of pure mathematics. Its relation to the latter is similar to that of mechanics orgeometry, if geometry is understood as a science of the properties of the real space. Nevertheless, apurely mathematical part can be isolated from it just as from geometry. For the latter, this was done atthe turn of the 19 th century, when it, considered as a part of pure mathematics, was transformed into ascience of a system of objects called points, straight lines, planes, and satisfying certain axioms.A similar full axiomatization of the theory of probability can be carried out by various methods.During the last years, the development of concrete branches of the theory was especially strongly