Point Process Convergence for Regularly Varying Sequences

In Sect. 6.1.4 we described extremal cluster sizes for univariate sequences \((X_t)\) with marginal distribution F by one number—the extremal index \(\theta _X\) —essentially the reciprocal of the expected cluster length. We discovered that the existence of a \(\theta _X\) smaller extremal index than 1 means that the limit distribution of the maxima in a dependent sample is downscaled when compared with an iid sample of the same size, and the distribution of \({\mathbb {P}}(M_n\le u_n)\) for high thresholds \(u_n\) essentially reduces to \((F(u_n))^{n\,\theta _X}\) .