Markov Kernels and the Conditional Extreme Value Model

The classical approach to extreme value modelling for multivariate data is to assume that the joint distribution belongs to a multivariate domain of attraction. In particular, this requires that each marginal distribution be individually attracted to a univariate extreme value distribution. The domain of attraction condition may be phrased conveniently in terms of regular variation of the joint distribution on an appropriate cone. A more flexible model for data realizations of a random vector was proposed by Heffernan and Tawn [45], under which not all the components are required to belong to an extremal domain of attraction. Such a model accomodates varying degrees of asymptotic dependence between pairs of components. Instead of starting from the joint distribution, assume the existence of an asymptotic approximation to the conditional distribution of the random vector given one of the components becomes extreme. Combined with the knowledge that the conditioning component belongs to a univariate domain of attraction, this leads to an approximation the probability of certain risk regions. When originally proposed, the focus was on conditional distributions. This approach presents technical difficulties regarding the choice of version but makes sense when dealing with Markov kernels. We place this approach in the more general approach using vague convergence of measures and multivariate regular variation on cones. MARKOV KERNELS AND THE CONDITIONAL EXTREME VALUE MODEL SIDNEY I. RESNICK AND DAVID ZEBER Abstract. The classical approach to extreme value modelling for multivariate data is to assume that the joint distribution belongs to a multivariate domain of attraction. In particular, this requires that each marginal distribution be individually attracted to a univariate extreme value distribution. The domain of attraction condition may be phrased conveniently in terms of regular variation of the joint distribution on an appropriate cone. A more flexible model for data realizations of a random vector was proposed by Heffernan and Tawn [45], under which not all the components are required to belong to an extremal domain of attraction. Such a model accomodates varying degrees of asymptotic dependence between pairs of components. Instead of starting from the joint distribution, assume the existence of an asymptotic approximation to the conditional distribution of the random vector given one of the components becomes extreme. Combined with the knowledge that the conditioning component belongs to a univariate domain of attraction, this leads to an approximation the probability of certain risk regions. When originally proposed, the focus was on conditional distributions. This approach presents technical difficulties regarding the choice of version but makes sense when dealing with Markov kernels. We place this approach in the more general approach using vague convergence of measures and multivariate regular variation on cones. The classical approach to extreme value modelling for multivariate data is to assume that the joint distribution belongs to a multivariate domain of attraction. In particular, this requires that each marginal distribution be individually attracted to a univariate extreme value distribution. The domain of attraction condition may be phrased conveniently in terms of regular variation of the joint distribution on an appropriate cone. A more flexible model for data realizations of a random vector was proposed by Heffernan and Tawn [45], under which not all the components are required to belong to an extremal domain of attraction. Such a model accomodates varying degrees of asymptotic dependence between pairs of components. Instead of starting from the joint distribution, assume the existence of an asymptotic approximation to the conditional distribution of the random vector given one of the components becomes extreme. Combined with the knowledge that the conditioning component belongs to a univariate domain of attraction, this leads to an approximation the probability of certain risk regions. When originally proposed, the focus was on conditional distributions. This approach presents technical difficulties regarding the choice of version but makes sense when dealing with Markov kernels. We place this approach in the more general approach using vague convergence of measures and multivariate regular variation on cones.