Asymptotic behavior of the moments of the ratio of the random sum of squares to the square of the random sum

Let {X1,X2,…} be a sequence of independent and identically distributed positive random variables of Pareto-type and let {N(t);t⩾0} be a mixed Poisson process independent of the Xi's. For any fixed t⩾0, define:TN(t)≔X12+X22+⋯+XN(t)2(X1+X2+⋯+XN(t))2if N(t)⩾1 and TN(t)≔0 otherwise. We determine the asymptotic behavior of any moment E[TN(t)k] as t→∞ with k∈N. Our method relies on the theory of functions of regular variation and an integral representation of these moments.