For stationary sequences X = {Xn}n≥1 we relate τ, the limiting mean number of exceedances of high levels un by X1,…,Xn, 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.

Empirical studies of data traffic in high-speed networks suggest that network traffic exhibits self-similarity and long-range dependence. Cumulative network traffic has been modeled using the so-called ON/OFF model. It was shown that cumulative network traffic can be approximated by either fractional Brownian motion or stable Lévy motion, depending on how many sources are active in the model. In this paper we consider exceedances of a high threshold by the sequence of lengths of ON-periods. If the cumulative network traffic converges to stable Lévy motion, the number of exceedances converges to a Poisson limit. The same holds in the fractional Brownian motion case, provided a very high threshold is used. Finally, we show that the number of exceedances obeys the central limit theorem.

Recent work by Athreya and Ney and by Nummelin on the limit theory for Markov chains shows that the close connection with regeneration theory holds also for chains on a general state space. Here this is used to study extremal behaviour of stationary (or asymptotically stationary) Markov chains. Many of the results center on the ‘clustering’ of extremes of adjacent values of the chains. In addition one criterion for convergence of extremes of general stationary sequences is derived. The results are applied to waiting times in the GI/G/1 queue and to autoregressive processes.