Asymptotics of sums of lognormal random variables with Gaussian copula

Abstract

Let (Y1, ..., Yn) have a joint n-dimensional Gaussian distribution with a general mean vector and a general covariance matrix, and let Xi = eYi, Sn = X1 + ⋯ + Xn. The asymptotics of P (Sn > x) as n → ∞ are shown to be the same as for the independent case with the same lognormal marginals. In particular, for identical marginals it holds that P (Sn > x) ∼ n P (X1 > x) no matter what the correlation structure is. © 2008 Elsevier B.V. All rights reserved.