Left tail of the sum of dependent positive random variables

We study the left tail behavior of the distribution function of a sum of dependent positive random variables, with a special focus on the setting of asymptotic independence. Asymptotics at the logarithmic scale are computed under the assumption that the marginal distribution functions decay slowly at zero, meaning that their logarithms are slowly varying functions. This includes parametric families such as log-normal, gamma, Weibull and many distributions from the financial mathematics literature. We show that the asymptotics of the sum 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, such as the Gaussian copula, the copulas of Gaussian mean-variance mixtures and a class of Archimedean copulas. As an illustration, we compute the left tail asymptotics for a portfolio of call options in the multidimensional Black-Scholes model.