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 be a mixed Poisson process independent of the Xi's. For any fixed t[greater-or-equal, slanted]0, define:if N(t)[greater-or-equal, slanted]1 and TN(t):=0 otherwise. We determine the asymptotic behavior of any moment as t-->[infinity] with . Our method relies on the theory of functions of regular variation and an integral representation of these moments.