For the sum of independent, subexponential random variables, second order asymptotic results for the tail are well studied in the literature. In the case that a mean exists, a heuristic interpretation of the second order approximation is that the sum is large if one summand is large and the others summands behave in a normal way, i.e. $n \mathbb P( X_1>u-(n-1) \mathbb E(X_1))$ is asymptotically a better approximation than the first order asymptotic  $ n \mathbb P( X_1>u)$.

In this talk we provide conditions under which similar results hold for dependent random variables. Further, for the example of multivariate lognormal random variables we discuss how this results can be used to construct efficient Monte Carlo estimators.