Tail Behaviour of Weighted Sums of Order Statistics of Dependent Risks

TL;DR

This paper derives the tail asymptotic behavior of weighted sums of order statistics for dependent risks, extending previous work to more general tail conditions and applications in risk theory.

Contribution

It provides a general tail asymptotic expansion for weighted sums of order statistics of dependent risks under broad tail conditions, including long-tailed and rapidly varying distributions.

Findings

Derived tail asymptotic expansion for weighted sums of order statistics.

Extended previous results to dependent risks with general tail behaviors.

Presented applications in risk theory demonstrating practical relevance.

Abstract

Let $X_{1},\ldots ,X_{n}$ 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 $X_{1:n}\leq \cdots \leq X_{n:n}$ of $X_{1},\ldots ,X_{n}$ under the general case in which the distribution function of $X_{n:n}$ is long-tailed or rapidly varying and $% X_{1},\ldots ,X_{n}$ may not be comparable in terms of their tail probability. We also present two examples and an application of our results in risk theory.