Classical extreme value theory for stationary sequences \(X_1,X_2,\dots\) of random variables can be paraphrased to a large extent as the study of exceedances over a high threshold \(u_\alpha\). A special role within the description of the temporal dependence between such exceedances is played by the extremal index \(\theta\in[0,1]\): For every \(\tau\in(0,\infty)\) there exists a sequence of thresholds \(u_n\), \(n=1,2,\dots\), such that \(nP(X_1>u_n)\to\tau\) and \(P\left(\max_{1\leq i\leq n}X_i\leq u_n\right)\to\exp(-\theta\tau)\) as \(n\to\infty\). Parts of this theory can be generalized not only to random variables on an arbitrary state space hitting certain failure sets, but even to a triangular array of rare events on an abstract probability space. In the case of M4 (maximum of multivariate moving maxima) processes, the arguments take a simple and direct form.