Randomly weighted sums under a wide type of dependence structure with application to conditional tail expectation

ABSTRACT

Let X_k, 1 ⩽ k ⩽ n, be n real-valued random variables, and θ_k, 1 ⩽ k ⩽ n, be another n non negative not-degenerate at zero random variables. Assume that random pairs (X₁, θ₁), …, (X_n, θ_n) are mutually independent, while each pair (X_k, θ_k) follows a wide type of dependence structure. Consider the randomly weighted sum S^θ_n = ∑_{k = 1}ⁿθ_kX_k. In this paper, the tail asymptotics for S^θ_n in the case where X_k, 1 ⩽ k ⩽ n, belong to some heavy-tailed subclasses are firstly investigated. Then, as an application, we consider the tail behavior of the conditional tail expectation as q↑1, where . Under some technical conditions, the asymptotic estimate for the right tail of conditional tail expectation is also derived. The obtained results extend some existing ones in the literature.

Keywords:

• Asymptotic
• conditional tail expectation
• heavy-tailed distribution
• randomly weighted sum.

Acknowledgments

The authors would like to thank the anonymous referees for their insightful and constructive suggestions which have helped us significantly improve the paper. The authors would also like to thank Dr. Hongyan Fang for providing the simulations.

Additional information

Funding

This work was supported by the Natural Science Foundation of Anhui Province (1808085MA16), the Provincial Natural Science Research Project of Anhui Colleges (KJ2017A024, KJ2017A028).