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 and Jef J. Teugels

Abstract: 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.

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

Classification (MSC2000): 60F05; 91B30

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