Strong mixing properties of max-infinitely divisible random fields

Let eta = (eta(t))(t is an element of T) be a sample continuous max-infinitely random field on a locally compact metric space T. For a closed subset S subset of T, we denote by eta(S) the restriction of eta to S. We consider beta(S-1, S-2) the absolute regularity coefficient between eta(S1) and eta(S2), 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) <= 2 integral P[eta not less than(S1) f, eta not less than(S2) f]mu(df), where f not less than(S) g means that there exists s is an element of S such that f(s) >= 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) <= 4 Sigma((s1,s2) is an element of S1 x S2) (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 Z(d). In the stationary max-stable case, we derive the asymptotic normality of three simple estimators of the pair extremal coefficient. (C) 2012 Elsevier B.V. All rights reserved.