Tail Behaviour of Weighted Sums of Order Statistics of Dependent Risks

Abstract

Let $X_1,\dots ,X_n$ be $n$ real-valued dependent random variables. With motivation from Mitra and Resnick (2009), we derive the tail asymptotic expansion for the weighted sum of order statistics $X_{1:n}\le \cdots \le X_{n:n}$ of $X_1,\dots ,X_n$ under the general case in which the distribution function of $X_{n:n}$ is long-tailed or rapidly varying and may not be comparable in terms of their tail probability. We also present two examples and an application of our results in risk theory.

References in corpus (1)

• Efficient simulation of tail probabilities for sums of log-elliptical risks