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

TL;DR

This paper investigates the clustering behavior of extremes in regularly-varying stochastic processes, revealing new phase transitions and convergence results that deepen understanding of extremal indices and rare event probabilities.

Contribution

It establishes convergence of point processes for extreme clusters and uncovers a phase transition in exceedance probabilities at the mesoscopic scale.

Findings

Convergence of point processes for extreme clusters.

Identification of a phase transition in exceedance probabilities.

Discrepancy between candidate and actual extremal indices explained.

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.