  We study the tail behavior of the distribution of the sum of asymptotically
independent risks whose marginal distributions belong to the maximal domain of
attraction of the Gumbel distribution. We impose conditions on the distribution
of the risks $(X,Y)$ such that $P(X + Y > x) \sim (const)P (X > x)$. With the
further assumption of non-negativity of the risks, the result is extended to
more than two risks. We note a sufficient condition for a distribution to
belong to both the maximal domain of attraction of the Gumbel distribution and
the subexponential class. We provide examples of distributions which satisfy
our assumptions. The examples include cases where the marginal distributions of
$X$ and $Y$ are subexponential and also cases where they are not. In addition,
the asymptotic behavior of linear combinations of such risks with positive
coefficients is explored leading to an approximate solution of an optimization
problem which is applied to portfolio design.
