Limit Distributions for the Ratio of the Random sum of Squares to the Square of the Random sum With Applications to Risk Measures

Sophie A. Ladoucette ; Jef J. Teugels

Publications de l'Institut Mathématique, _N_S_80 (2006) no. 94, p. 219

Résumé

Let $\{X_1,X_2,\ldots\}$ be a sequence of independent and identically distributed positive random variables of Pareto-type 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+\cdots+X_{N(t)}^2}{(X_1+X_2+\cdots+X_{N(t)})^2} $ if $N(t)\geq 1$ and $T_{N(t)}:=0$ otherwise. We derive limits in distribution for $T_{N(t)}$ under some convergence conditions on 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 $\mathbb{E}X_1^2=\infty$). Armed with these results, we obtain asymptotic properties of two popular risk measures, namely the sample coefficient of variation and the sample dispersion.

Détail

DOI : 10.2298/PIM0694219L

Classification : 60F05 91B30
Keywords: Counting process, Domain of attraction of a stable distribution, Functions of regular variation, Pareto-type distribution, Sample coefficient of variation, Sample dispersion, Weak convergence