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)∼(constant) P(X›x). With the further assumption of
nonnegativity 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.
Primary Subjects: 60E07, 60G51, 60G52, 60G70
Secondary Subjects: 60F17, 62G30, 90B50, 90C59
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