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 ∞ λ n ( η n − E η n ) , where η n are independent gamma random variables with shape parameters r n > 0 , and the λ n ’s are predetermined weights. We study the asymptotic behavior of the tail ∑ n = M ∞ λ n ( η n − E η n ) , 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. sults we obtain are directly applicable to the study of double Wiener–Itô integrals and to the “Rosenblatt distribution”.