  In many areas of interest, modern risk assessment requires estimation of the
extremal behaviour of sums of random variables. We derive the first order
upper-tail behaviour of the weighted sum of bivariate random variables under
weak assumptions on their marginal distributions and their copula. The extremal
behaviour of the marginal variables is characterised by the generalised Pareto
distribution and their extremal dependence through subclasses of the limiting
representations of Ledford and Tawn (1997) and Heffernan and Tawn (2004). We
find that the upper tail behaviour of the aggregate is driven by different
factors dependent on the signs of the marginal shape parameters; if they are
both negative, the extremal behaviour of the aggregate is determined by both
marginal shape parameters and the coefficient of asymptotic independence
(Ledford and Tawn, 1996); if they are both positive or have different signs,
the upper-tail behaviour of the aggregate is given solely by the largest
marginal shape. We also derive the aggregate upper-tail behaviour for some well
known copulae which reveals further insight into the tail structure when the
copula falls outside the conditions for the subclasses of the limiting
dependence representations.
