Some theorems on the sum of positive random variables

Abstract

The purpose of this paper is to treat the limiting distribution and the fluctuation of the sum of identically distributed positive random variables without mean, which frequently appear in the theory of recurrence time. Let X~, X2, . . , Xn be positive random variables without mean, and F(x) be their common distribution function, and let C, be such that 1-F(Cn)--~ 1_ (for large n). Then the sum n