Integrability of infinite weighted sums of heavy-tailed i.i.d. random variables

We consider the sum X of i.i.d. random variables Y-n, n greater than or equal to 0, with weights a(n), which decay exponentially fast to zero. For a smooth sublinear increasing function g, g(\Y-0\) has finite expectation if and only if the expectation of \X\g'(\X\) is finite. The proof uses characteristic functions. However, if g grows polynomially or exponentially fast, then the expectation of g(\Y-0\) is finite if and only if the expectation of g(\X\) is finite. (C) 2002 Elsevier Science B.V. All rights reserved.