Max-semistable extreme value laws for autoregressive processes with Cantor-like marginals

TL;DR

This paper studies autoregressive processes with Cantor-like marginals, showing their extreme values follow a max-semistable law influenced by an extremal index, linking stochastic processes with dynamical systems.

Contribution

It introduces max-semistable extreme value laws for autoregressive processes with Cantor-like marginals, connecting extreme value theory with dynamical systems.

Findings

Marginal distributions are in the domain of attraction of max-semistable laws.

Extreme value law incorporates an extremal index similar to i.i.d. cases.

Connections established between stochastic processes and deterministic dynamical systems.

Abstract

This paper considers a family of autoregressive processes with marginal distributions resembling the Cantor function. It is shown that the marginal distribution is in the domain of attraction of a max-semistable distribution. The main result is that the extreme value law for the autoregressive process is obtained by including an extremal index in the law for an i.i.d.\ process with the same marginal distribution. Connections with extremes in deterministic dynamical systems and the relevance of max-semistable distributions in that context are also pointed out.