Strong mixing properties of max-infinitely divisible random fields

Let η = ( η ( t ) ) t ∈ T be a sample continuous max-infinitely random field on a locally compact metric space T . For a closed subset S ⊂ T , we denote by η S the restriction of η to S . We consider β ( S 1 , S 2 ) , the absolute regularity coefficient between η S 1 and η S 2 , where S 1 , S 2 are two disjoint closed subsets of T . Our main result is a simple upper bound for β ( S 1 , S 2 ) involving the exponent measure μ of η : we prove that β ( S 1 , S 2 ) ≤ 2 ∫ P [ η ≮ S 1 f , η ≮ S 2 f ] μ ( d f ) , where f ≮ S g means that there exists s ∈ S such that f ( s ) ≥ g ( s ) . s 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 β ( S 1 , S 2 ) ≤ 4 ∑ ( s 1 , s 2 ) ∈ S 1 × S 2 ( 2 − θ ( s 1 , s 2 ) ) , where θ ( s 1 , s 2 ) is the pair extremal coefficient. 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.