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

Abstract

Abstract. Let {X 1 , X 2 , . . .} 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 X i 's. For any fixed t 0, define:if N (t) 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 (EX 1 = ∞) or when only the numerator has an erratic behavior (EX 1 < ∞ and EX 2 1 = ∞). Armed with these results, we obtain asymptotic properties of two popular risk measures, namely the sample coefficient of variation and the sample dispersion.

Domains of attraction of the real random vector (X,X2) and applications

“…Using the notations of Theorem 2.3 and Remark 2.3, we have the following result. The result completes the proofs of the corresponding results of Albrecher and Teugels (2004) or Ladoucette and Teugels (2006), Ladoucette (2007), Omey (2008a). In Theorem 3.1 below we consider the case where ̸ = 1.…”

“…In Albrecher and 2000 Mathematics Subject Classification: Primary 60E05, 60F05; Secondary 62E20, 91B70. Teugels (2004) and Ladoucette and Teugels (2006), the authors discuss asymptotic properties of ( ) and ( ). In the case where 0, Omey (2008a) obtained a detailed and complete analysis of ( ), ( ) and ( ).…”

Asymptotic behavior of the moments of the ratio of the random sum of squares to the square of the random sum

“…Ladoucette and Teugels [8] focus on weak convergence by deriving limit distributions for the appropriately normalized ratio T N (t) as t → ∞ when X is of Pareto-type with index α > 0 and the counting process {N (t); t ≥ 0} satisfies some convergence conditions according to the range of α. Armed with their results on T N (t) and thanks to the relation (10), they also derive asymptotic properties of the sample coefficient of variation, even when the first moment and/or the second moment of X do not exist.…”

“…Armed with their results on T N (t) and thanks to the relation (10), they also derive asymptotic properties of the sample coefficient of variation, even when the first moment and/or the second moment of X do not exist. Furthermore, Ladoucette and Teugels [8] adapt the methodology to derive asymptotic properties of another measure of variation, namely the sample dispersion. Recall that the value of the dispersion allows to compare the volatility with respect to the Poisson case.…”

Analysis of risk measures for reinsurance layers

“…For the asymptotic behavior of the sample coefficient of variation as well as for the sample dispersion, see the recent papers by Ladoucette (2005) and Ladoucette and Teugels (2005).…”