Let Xi, i = 1,…, n, be a sequence of positive independent identically distributed random variables and define (Formula Presented). Utilizing Karamata’s theory of functions of regular variation, we determine the asymptotic behaviour of arbitrary moments (Formula Presented), k ∈ ℕ, for large n, given that X1 satisfies a tail condition, akin to the domain of attraction condition from extreme value theory. As a by-product, the paper offers a new method for estimating the extreme value index of Pareto-type tails.