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

We consider the sum X of i.i.d. random variables Yn, n[greater-or-equal, slanted]0, with weights an which decay exponentially fast to zero. For a smooth sublinear increasing function g, g(Y0) has finite expectation if and only if the expectation of Xg'(X) is finite. The proof uses characteristic functions. However, if g grows polynomially or exponentially fast, then the expectation of g(Y0) is finite if and only if the expectation of g(X) is finite.