On the Tail Behaviour of Aggregated Random Variables

Modern risk assessment in many areas of interest require 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 is characterised using subclasses of the limiting representations of Ledford and Tawn (1997) and Heffernan and Tawn (2004), which describe both components being jointly extreme, and the behaviour of one component conditional on the other component being large, respectively. We find that the upper are both positive or have different signs, 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 driven 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 driven solely be 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. We conduct inference on the upper-tail of aggregates of gridded UK precipitation and temperature data to illustrate that our limit results provide good approximations in practice.