A remarkable example on clustering of extremes for regularly-varying stochastic processes

Abstract

The stable-regenerative multiple-stable model has been shown recently to have distinct candidate extremal index and extremal index. To understand further this rare phenomenon, two more results are established here for the double-stable model. The first is the convergence of point processes for the clusters of extremes, enhancing the previous result on the weak convergence of random sup-measures. Most interestingly, the second result reveals a new phase transition at the mesoscopic level when computing the asymptotic exceedance probability over a block, $\mathbb{P}({\max }_{k=1,\dots ,d_n}{X}_k>b_n)$, as $n\to \infty$. Here, the mesoscopic level is referred to the fact that the block size $d_n$ is allowed to grow at the rate $n^{\rho }$ with $\rho \in [0,1]$, while the threshold $b_n$ is such that $\mathbb{P}(X_1>b_n)\sim 1/n$. The recently discovered discrepancy between the candidate extremal index and the extremal index is shown to be just a reflection of this phase transition that is prohibited by the anticlustering condition.

38 pages; corrected a few typos and implemented some minor edits