Let \{ X_1, X_2, ...\} be a sequence of positive independent and identically distributed random variables of Pareto-type with index \alpha>0 and let \{ N(t); t\geq 0\} be a mixed Poisson process independent of the X_i's. For t\geq 0, define T_{ N(t)}:=\frac{ X_1^2 + X_2^2 + ... + X_{ N(t)}^2} { (X_1 + X_2 + ... + X_{ N(t)})^2} if N(t)\geq 1 and T_{ N(t)}:=0 otherwise. We derive the limiting behavior of the k-th moment of T_{ N(t)}, k\in\mathbb{ N}, by using the theory of functions of regular variation and an integral representation for \mathbb{ E}\{ T_{ N(t)}^k\}. We also point out the connection between T_{ N(t)} and the sample coefficient of variation which is a popular risk measure in practical applications.
Categories: math.PR math.ST stat.TH