Strong mixing properties of max-infinitely divisible random fields

TL;DR

This paper establishes bounds on the mixing properties of max-infinitely divisible random fields using the exponent measure, linking extremal coefficients to mixing, and applies these results to prove a central limit theorem and estimator asymptotics.

Contribution

It provides a new upper bound for the absolute regularity coefficient of max-infinitely divisible fields based on the exponent measure, connecting extremal coefficients to mixing properties.

Findings

Derived an upper bound for the absolute regularity coefficient involving the exponent measure.

Linked extremal coefficients to mixing bounds in max-stable fields.

Proved a central limit theorem for stationary max-infinitely divisible fields and asymptotic normality of estimators.

Abstract

Let $\eta=(\eta(t))_{t\in T}$ be a sample continuous max-infinitely random field on a locally compact metric space $T$. For a closed subset $S\in T$, we note $\eta_{S}$ the restriction of $\eta$ to $S$. We consider $\beta(S_1,S_2)$ the absolute regularity coefficient between $\eta_{S_1}$ and $\eta_{S_2}$, where $S_1,S_2$ are two disjoint closed subsets of $T$. Our main result is a simple upper bound for $\beta(S_1,S_2)$ involving the exponent measure $\mu$ of $\eta$: we prove that $\beta(S_1,S_2)\leq 2\int \bbP[\eta\not<_{S_1} f,\ \eta\not <_{S_2} f]\,\mu(df)$, where $f\not<_{S} g$ means that there exists $s\in S$ such that $f(s)\geq g(s)$. If $\eta$ is a simple max-stable random field, the upper bound is related to the so-called extremal coefficients: for countable disjoint sets $S_1$ and $S_2$, we obtain $\beta(S_1,S_2)\leq 4\sum_{(s_1,s_2)\in S_1\times S_2}(2-\theta(s_1,s_2))$, where $\theta(s_1,s_2)$ is the pair extremal coefficient. As an application, we show that these new estimates entail a central limit theorem for stationary max-infinitely divisible random fields on $\bbZ^d$. In the stationary max-stable case, we derive the asymptotic normality of three simple estimators of the pair extremal coefficient.