Asymptotic results on the moments of the ratio of the random sum of squares to the square of the random sum

Abstract

Let \{X_1, X_2, ...\} be a sequence of positive independent and identically distributed random variables of Pareto-type with index α>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.