Berry–Esseen and Edgeworth approximations for the normalized tail of an infinite sum of independent weighted gamma random variables

Consider the sum Z=∑n=1∞λn(ηn−Eηn), where ηnηn are independent gamma random variables with shape parameters rn>0rn>0, and the λnλn’s are predetermined weights. We study the asymptotic behavior of the tail ∑n=M∞λn(ηn−Eηn), which is asymptotically normal under certain conditions. We derive a Berry–Esseen bound and Edgeworth expansions for its distribution function. We illustrate the effectiveness of these expansions on an infinite sum of weighted chi-squared distributions.The results we obtain are directly applicable to the study of double Wiener–Itô integrals and to the “Rosenblatt distribution”.