Limit conditional distributions for bivariate vectors with polar representation

Anne-Laure Fougères (1) , Philippe Soulier (2)

1 ICJ - Institut Camille Jordan
2 MODAL'X - Modélisation aléatoire de Paris X

We investigate conditions for the existence of the limiting conditional distribution of a bivariate random vector when one component becomes large. We revisit the existing literature on the topic, and present some new sufficient conditions. We concentrate on the case where the conditioning variable belongs to the maximum domain of attraction of the Gumbel law, and we study geometric conditions on the joint distribution of the vector. We show that these conditions are of a local nature and imply asymptotic independence when both variables belong to the domain of attraction of an extreme value distribution. The new model we introduce can also be useful for simulations.

Mots clés: second order correction; Conditional excess probability; conditional extreme-value model; $\Gamma$-varying tail; asymptotic independence; elliptic distributions; second order correction. Conditional excess probability; conditional extreme-value mo\-del; second order correction.

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Théorie [stat.TH]

Anne-Laure Fougères, Philippe Soulier. Limit conditional distributions for bivariate vectors with polar representation. Stocahstic models, 2010, 26 (1), pp.54-77. ⟨10.1080/15326340903291362⟩. ⟨hal-00373049v2⟩