The upcrossings index and the extremal index

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 conditions D̃ (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.