Asymptotic analysis for the ratio of the random sum of squares to the square of the random sum with applications to risk measures

Let {X_1, X_2, …} be a sequence of independent and identically distributed positive random variables of Pareto-type with index \alpha>0 and let {N(t); t\geq 0} be a counting process independent of the X_i’s. For any fixed 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 limiting distributions for T_{N(t)} by assuming some convergence properties for the counting process. This is even achieved when both the numerator and the denominator defining T_{N(t)} exhibit an erratic behavior ($\mathbb{E}$X_1=\infty) or when only the numerator has an erratic behavior ($\mathbb{E}$X_1<\infty and $\mathbb{E}$X_1^2=\infty). Thanks to these results, we obtain asymptotic properties pertaining to both the sample coefficient of variation and the sample dispersion.