We study the left tail behavior of the logarithm of the distribution function of a sum of dependent positive random variables . Asymptotics are computed under the assumption that the marginal distribution functions decay slowly at zero, meaning that the 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 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 . As an illustration, we compute the left tail asymptotics for a portfolio of  options in the multidimensional Black-Scholes model.
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.
D
logarithm of the
R
. Asymptotics
, with a special focus on the setting of asymptotic independence. Asymptotics at the logarithmic scale
D
the
D
logarithmic
D
in question
R
, and which is computed
. We then compute this function
R
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
, such as the Gaussian copula, the copulas of Gaussian mean-variance mixtures and a class of Archimedean copulas
A
call