Bikramjit Das and Sidney Resnick

Multivariate extreme value theory assumes a multivariate domain of attraction condition for the distribution of a random vector necessitating that each component satisfy a marginal domain of attraction condition. Heffernan and Tawn (2004) and Heffernan and Resnick (2007) developed an approximation to the joint distribution of the random vector by conditioning that one of the components be extreme. Prior papers left unresolved the consistency of different models obtained by conditioning on different components being extreme and we provide understanding of this issue. We also clarify the relationship between the conditional distributions and multivariate extreme value theory. We discuss conditions under which the two models are the same and when one can extend the conditional model to the extreme value model. We also discuss the relationship between the conditional extreme value model and standard regular variation on cones of the form $[0,\infty]\times(0,\infty]$ or $(0,\infty]\times[0,\infty]$.