Limit distributions for the ratio of the random sum of squares to the square of the random sum with applications to risk measures

Abstract

Let {X1,X2, . . .} be a sequence of independent and identically distributed positive random variables of Pareto-type and let {N(t); t >_ 0} be a counting process independent of the Xi?s. For any fixed t> _ 0, define: TN(t) := X2 1 + X2 2 + ? ? ? + X2N (t) (X1 + X2 + ? ? ? + XN(t))2 if N(t) >_ 1 and TN(t) := 0 otherwise. We derive limits in distribution for TN(t) under some convergence conditions on the counting process. This is even achieved when both the numerator and the denominator defining TN(t) exhibit an erratic behavior (EX1 = ?) or when only the numerator has an erratic behavior (EX1 < ? and EX2 1 = ?). Armed with these results, we obtain asymptotic properties of two popular risk measures, namely the sample coefficient of variation and the sample dispersion. References.