Limit conditional distributions for bivariate vectors with polar representation

TL;DR

This paper explores conditions under which the limiting conditional distribution of a bivariate vector exists as one component becomes large, focusing on Gumbel domain cases and geometric conditions, with implications for asymptotic independence and simulation.

Contribution

It introduces new sufficient conditions for the existence of limiting conditional distributions in bivariate vectors, especially in the Gumbel domain, and discusses geometric criteria related to asymptotic independence.

Findings

Conditions are of a local nature and imply asymptotic independence.

New geometric conditions on joint distribution are established.

The model facilitates simulations of extreme events.

Abstract

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.