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⩾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 |X|g′(|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.