On the Tail Behaviour of Aggregated Random Variables

Richards, Jordan and Tawn, Jonathan (2022) On the Tail Behaviour of Aggregated Random Variables. Journal of Multivariate Analysis, 192. ISSN 0047-259X

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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 and Heffernan and Tawn. 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; 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.

Item Type:
Journal Article
Journal or Publication Title:
Journal of Multivariate Analysis
Additional Information:
This is the author’s version of a work that was accepted for publication in Journal of Multivariate Analysis. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Multivariate Analysis, 192, 105065, 2022 DOI: 10.1016/j.jmva.2022.105065
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10 Jun 2022 11:45
Last Modified:
22 Sep 2023 00:51