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1. Notation and generalities. We shall in this paper deal with sums of independent, identically distributed, positive, continuous random variables, whose distribution lies within the domain of attraction of a stable law. We denote the random variables by Xk, k = 1, 2, * , their distribution function by F(x), and their sum by Sn = 1 Xk- We thus are given (1) P(Xk > x) = 1 -F(x) = h(x)lxy for x > xo where 0 <y <2, and h(x) is such that for any positive constant c, rim h(cn)/h(n) = 1.

On strong bounds for sums of independent random variables which tend to a stable distribution