Functionals of clusters of extremes

For arbitrary stationary sequences of random variables satisfying a mild mixing condition, distributional approximations are established for functionals of clusters of exceedances over a high threshold. The approximations are in terms of the distribution of the process conditionally on the event that the first variable exceeds the threshold. This conditional distribution is shown to converge to a nontrivial limit if the finite-dimensional distributions of the process are in the domain of attraction of a multivariate extreme-value distribution. In this case, therefore, limit distributions are obtained for functionals of clusters of extremes, thereby generalizing results for higher-order stationary Markov chains by Yun (2000).

Functionals of clusters of extremes

For arbitrary stationary sequences of random variables satisfying a mild mixing condition, distributional approximations are established for functionals of clusters of exceedances over a high threshold. The approximations are in terms of the distribution of the process conditionally on the event that the first variable exceeds the threshold. This conditional distribution is shown to converge to a nontrivial limit if the finite-dimensional distributions of the process are in the domain of attraction of a multivariate extreme-value distribution. In this case, therefore, limit distributions are obtained for functionals of clusters of extremes, thereby generalizing results for higher-order stationary Markov chains by Yun (2000).