Distribution of sums of independent random variables

Abstract

i. Introductory Remarks and Statement of Results We will consider a sequence of independent random variables 41, ~.~ . . . . . ~ . . . . . . (1) with zero mathematical expectation and distribution function Fx (x), F. (x) . . . . . F. (x) . . . . satisfying the condition O ~ k = ~ ~0, b1~>O, a1+b1=l, Vx(x) is a continuous distribution function with density Vx(x) = v(x), and V2(x) is a distribution function consisting only of a singular and purely stepwise component. This decomposition can be altered and we obtain the decomposition F (:c)= a Q (x) + bT(x) , in which the distribution function Q(x) has bounded density. In this decomposition we also havea~>0, b~>0, and a + b = i. But T(x) will be a distribution function that may have a singular, stepwise, as well as continuous components. Such a decomposition can be obtained at least in the following way. We represent the distribution function P(x) in the form of a sum of nonnegative functions, F (x) = F1 (x) + F2 (x), where F,'(x) = p(x) exists. We select a constant Cx and construct the two functions