Phase transition for extremes of a family of stationary multiple-stable processes

Abstract

We investigate a family of multiple-stable processes that may exhibit either long-range or short-range dependence, depending on the parameters. There are two parameters for the processes, the memory parameter $\beta \in (0,1)$ and the multiplicity parameter $p\in \mathbb{N}$. We investigate the macroscopic limit of extremes of the process, in terms of convergence of random sup-measures, for the full range of parameters. Our results show that (i) the extremes of the process exhibit long-range dependence when ${\beta }_p:=p\beta -p+1\in (0,1)$, with a new family of random sup-measures arising in the limit, (ii) the extremes are of short-range dependence when ${\beta }_p<0$, with independently scattered random sup-measures arising in the limit, and (iii) there is a delicate phase transition at the critical regime ${\beta }_p=0$.

Major revision. 36 pages

References in corpus (2)

• A Berry-Esseén theorem for partial sums of functionals of heavy-tailed moving averages
• Tail processes for stable-regenerative multiple-stable model