Ladoucette, Sophie A.; Teugels, Jef J. Limit distributions for the ratio of the random sum of squares to the square of the random sum with applications to risk measures. (English) Zbl 1164.60328 Publ. Inst. Math., Nouv. Sér. 80(94), 219-240 (2006). Let \(\{X_n,\,n\geq 1\}\) be a sequence of iid positive random variables having regularly varying tails with index \(-\alpha<0\), let \(\{N(t);\,t\geq 0\}\) be a counting process independent of the \(X_i\)’s and, 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. Under the appropriate conditions on the counting process, the authors derive weak limits for \(T_{N(t)}\). These results are used in the second part of this paper to obtain the asymptotic behavior of two risk measures: the sample coefficient of variation and the sample dispersion. Reviewer: Slobodanka Janković (Beograd) Cited in 3 Documents MSC: 60F05 Central limit and other weak theorems 91B30 Risk theory, insurance (MSC2010) 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 PDFBibTeX XMLCite \textit{S. A. Ladoucette} and \textit{J. J. Teugels}, Publ. Inst. Math., Nouv. Sér. 80(94), 219--240 (2006; Zbl 1164.60328) Full Text: DOI EuDML