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.
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 . We then compute this function explicitly for several families of copulas in this paper.