Strong mixing properties of max-infinitely divisible random fields

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.