Tails of higher-order moments of sums with heavy-tailed increments and application to the Haezendonck–Goovaerts risk measure

Abstract

In this paper we consider the sum Snξ≔ξ1+…+ξn of (possibly dependent and nonidentically distributed) real-valued random variables ξ1,…,ξn with dominatedly varying distributions. Assuming that the ξk’s follow the dependence structure, similar to the asymptotic independence, we obtain the asymptotic lower and upper bounds for the tail moment E((Snξ)m1{Snξ>x}), where m is a nonnegative integer, improving the bounds of Leipus et al. (2019). We also consider the case of nonnegative random variables. Using the obtained results, we get the asymptotic estimations for the Haezendonck–Goovaerts risk measure in two examples of sums with regularly varying and dominatedly varying (but not regularly varying) increments.