Strong mixing properties of max-infinitely divisible random fields

Let η=(η(t))t∈Tη=(η(t))t∈T be a sample continuous max-infinitely random field on a locally compact metric space TT. For a closed subset S⊂TS⊂T, we denote by ηSηS the restriction of ηη to SS. We consider β(S1,S2)β(S1,S2), the absolute regularity coefficient between ηS1ηS1 and ηS2ηS2, where S1,S2S1,S2 are two disjoint closed subsets of TT. Our main result is a simple upper bound for β(S1,S2)β(S1,S2) involving the exponent measure μμ of ηη: we prove that β(S1,S2)≤2∫P[η≮S1f,η≮S2f]μ(df), where f≮Sgf≮Sg means that there exists s∈Ss∈S such that f(s)≥g(s)f(s)≥g(s).If ηη is a simple max-stable random field, the upper bound is related to the so-called extremal coefficients: for countable disjoint sets S1S1 and S2S2, we obtain β(S1,S2)≤4∑(s1,s2)∈S1×S2(2−θ(s1,s2))β(S1,S2)≤4∑(s1,s2)∈S1×S2(2−θ(s1,s2)), where θ(s1,s2)θ(s1,s2) 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 ZdZd. In the stationary max-stable case, we derive the asymptotic normality of three simple estimators of the pair extremal coefficient.