  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.
