  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.
