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

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, P(maxk=1,…,dnXk>bn), as n→∞. Here, the mesoscopic level is referred to the fact that the block size dn is allowed to grow at the rate nρ with ρ∈(0,1), while the threshold bn is such that P(X1>bn)∼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.