Asymptotic analysis for the ratio of the random sum of squares to the square of the random sum with applications to risk measures

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dc.creatorLadoucette, S. A.
dc.creatorTeugels, J. L.
dc.date2005-11-03
dc.date.accessioned2026-07-07T08:07:21Z
dc.date.available2026-07-07T08:07:21Z
dc.descriptionLet \{X_1, X_2, ...\} be a sequence of independent and identically distributed positive random variables of Pareto-type with index α>0 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 + ... + X_{N(t)}^2} {(X_1 + X_2 + ... + X_{N(t)})^2} if N(t)\geq 1 and T_{N(t)}:=0 otherwise. We derive limiting distributions for T_{N(t)} by assuming some convergence properties for 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). Thanks to these results, we obtain asymptotic properties pertaining to both the sample coefficient of variation and the sample dispersion.
dc.description21 pages
dc.identifierhttps://arxiv.org/abs/math/0511082
dc.identifierhttp://arxiv.org/abs/math/0511082
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/130915
dc.subjectProbability
dc.subjectStatistics Theory
dc.subject60F05; 91B30
dc.titleAsymptotic analysis for the ratio of the random sum of squares to the square of the random sum with applications to risk measures
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