Before revision:  We show that the logarithmic asymptotics of the sum in question depend on a characteristic of the copula of the random variables which we term weak lower tail dependence function , and which is computed explicitly for several families of copulas in this paper. In applications, our results may be used to quantify the diversification of long-only portfolios of financial assets with respect to extreme losses .
After revision:  We show that the logarithmic asymptotics of the sum in question depend on a characteristic of the copula of the random variables which we term weak lower tail dependence function , and which is computed explicitly for several families of copulas , such as the Gaussian copula, the copulas of Gaussian mean-variance mixtures and a class of Archimedean copulas .