No abstract

For stationary sequences X = {X n } n≥1 we relate τ , the limiting mean number of exceedances of high levels u n by X 1 , . . . , X n , and ν, the limiting mean number of upcrossings of the same level, through the expression θ = (ν/τ )η, where θ is the extremal index of X and η is a new parameter here called the upcrossings index. The upcrossings index is a measure of the clustering of upcrossings of u by variables in X, and the above relation extends the known relation θ = ν/τ , which holds under the mild-oscillation local restriction D (u) on X. We present a new family of local mixing conditionsD (k) (u) under which we prove that (a) the intensity of the limiting point process of upcrossings and η can both be computed from the k-variate distributions of X; and (b) the cluster size distributions for the exceedances are asymptotically equivalent to those for the lengths of one run of exceedances or the lengths of several consecutive runs which are separated by at most k − 2 nonexceedances and, except for the last one, each contain at most k − 2 exceedances.

For stationary sequences, under general local and asymptotic dependence restrictions, any limiting point process for time normalized upcrossings of high levels is a compound Poisson process, i.e., there is a clustering of high upcrossings, where the underlying Poisson points represent cluster positions, and the multiplicities correspond to cluster sizes. For such classes of stationary sequences there exists the upcrossings index η, 0 ≤ η ≤ 1, which is directly related to the extremal index θ, 0 ≤ θ ≤ 1, for suitable high levels. In this paper we consider the problem of estimating the upcrossings index η for a class of stationary sequences satisfying a mild oscillation restriction. For the proposed estimator, properties such as consistency and asymptotic normality are studied. Finally, the performance of the estimator is assessed through simulation studies for autoregressive processes and case studies in the fields of environment and finance.

Due to globalization and relaxed market regulation, we have assisted to an increasing of extremal dependence in international markets. As a consequence, several measures of tail dependence have been stated in literature in recent years, based on multivariate extreme-value theory. In this paper we present a tail dependence function and an extremal coefficient of dependence between two random vectors that extend existing ones. We shall see that in weakening the usual required dependence allows to assess the amount of dependence in d-variate random vectors based on bidimensional techniques. Very simple estimators will be stated and can be applied to the well-known stable tail dependence function. Asymptotic normality and strong consistency will be derived too. An application to financial markets will be presented at the end.

We present local dependence conditions for stationary random fields under which the extremal index and the asymptotic distribution of the maximum M (n 1 ,...,n d ) can be calculated from the joint distribution of a finite number s 1 s 2 of variables.keywords: Extremal index, local and long range dependence, random field.