7. Probability and Statistics Soviet Essays - Sheynin, Oscar

and some quite definite (not dependent on randomness) function f(n). We shall determine the set ofsuch values of n for which n > f(n). If this set is finite with probability 1, we shall call f(n) the upperfunction of the sequence (1.7.1); if it is infinite, again with probability 1, f(n) will be the lowerfunction. According to the formulation of §1.6, the strong law of large numbers means nothing elsebut that for any > 0 the function f(n) = n is the upper function for the absolute values n = | n – E n |of the deviations of the sums n from their expectations. The desire to ascertain for eachsequence of independent random variables 1 , 2 , …, n , …, the class of the upper functions forthe corresponding absolute deviations n is quite natural.In 1924 Khinchin [2] provided an almost exhausting solution of this problem for the case of theBernoulli trials (when the variables n were identically distributed and only took two values). Hisfinding can be formulated thus: For any > 0f(n) = (1 + )2 Bnlog log Bnand g(n) = (1 – ) 2 Bnlog log Bnare the upper and lower functions respectively. Here, as above, we assume that B n = var n .Later Kolmogorov [4] discovered that the same law also takes place under considerably moregeneral conditions. After that the problem was mostly considered by foreign authors. Among theeasily formulated findings we should indicate that the law of the iterated logarithm is in any casevalid for identically distributed terms n under a single (unavoidable according to the veryformulation of the law) restriction, the finiteness of their variances.The conditions for the application of the law of the iterated logarithm are not yet exhaustivelystudied. Various foreign authors have expended much efforts on widening these conditions as well ason a more precise separation of the upper functions from the lower ones. The main fundamentalprogress was, however, achieved here in the Soviet Union by Petrovsky [2]. He himself only solved aproblem to which Kolmogorov’s abovementioned methods led and which belonged to the theory ofdifferential equations. Petrovsky’s research enables to formulate the following test: Supposing that(t) is a monotone function, f(n) = (B n ) will be the upper function if the integral ∞0(1/t) (t) exp[– 2 (t)/2]dtconverges, and the lower if it diverges. Certain conditions justifying the transition to Petrovsky’sproblem are still required for applying his criterion; Erdös and Feller, see Feller (1943), ascertainedthem.2. Sums of Weakly Dependent Random VariablesThe main properties of sums of independent terms persist if the dependence between the terms issufficiently weak, or sufficiently rapidly weakens when the difference between their serial numbersincreases. In the latter case, the terms are supposed to be enumerated in some definite natural order.This idea, first put forward and worked out by Markov, was further developed in a number of Sovietinvestigations.2.1. The Law of Large Numbers. If all the terms of the sum (1.1.6) have finite variances, then thevariance B n = var n can be written down asor asnB n =i = 1nj = 1c ij (n) (2.1.1)