Before revision: 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.
After revision: We study the left tail behavior of the logarithm 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 the their logarithms are slowly varying functions.