Course in Probability Theory

25 0 1 CENTRAL LIMIT THEOREM AND ITS RAMIFICATIONSShow that for sufficiently large k the event ek n f k implies the complementof ek+1 ; hence deducek+1 k~~ n e j < ~(elo) 11[1 - ow j )]j=joj=joand show that the product -k 0 as k -3 00 .7 .6 Infinite divisibilityThe weak law of large numbers and the central limit theorem are concerned,respectively, with the convergence in dist . of sums of independent r .v .'s to adegenerate and a normal d .f. It may seem strange that we should be so muchoccupied with these two apparently unrelated distributions . Let us point out,however, that in terms of ch .f.'s these two may be denoted, respectively, byeat and e°`t-b't2 - exponentials of polynomials of the first and second degreein (it) . This explains the considerable similarity between the two cases, asevidenced particularly in Theorems 6 .4 .3 and 6 .4 .4 .Now the question arises : what other limiting d .f.'s are there when smallindependent r .v.'s are added? Specifically, consider the double array (2) inSec . 7 .1, in which independence in each row and holospoudicity are assumed .Suppose that for some sequence of constants a n ,e~S n- aj=1converges in dist . to F . What is the class of such F's, and when doessuch a convergence take place? For a single sequence of independent r .v .'s{X j , j > 1}, similar questions may be posed for the "normed sums" (S 7z -a n )/b n .These questions have been answered completely by the work of Levy, Khintchine,Kolmogorov, and others ; for a comprehensive treatment we refer to thebook by Gnedenko and Kolmogorov [12] . Here we must content ourselveswith a modest introduction to this important and beautiful subject .We begin by recalling other cases of the above-mentioned limiting distributions,conveniently dispilayed by their ch .f.'s :i.>0 ; e0