Asymptotic analysis of a measure of variation

Let $X_i$, $i=1,\dots ,n$, be a sequence of positive independent identically distributed random variables and define \[ T_n:=\frac {X_1^2+X_2^2+\dots +X_n^2}{(X_1+X_2+\dots +X_n)^2}. \] Utilizing Karamata’s theory of functions of regular variation, we determine the asymptotic behaviour of arbitrary moments $\mathsf {E}(T_n^k)$, $k\in \mathbb {N}$, for large $n$, given that $X_1$ 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.