On maxima of stationary fields

2018/10/10 by Soja-Kukieła, Natalia
#60G60 #60G70 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1810.04496

Abstract

Let \X_n : n∈ℤ^d\ be a weakly dependent stationary field with maxima M_A := sup\X_i : i∈ A\ for finite A⊂ℤ^d and M_n := sup\X_i : 1 ≤ i ≤ n \ for n∈ℕ^d. In a general setting we prove that P(M_{(n,n,…, n)} ≤ v_n) = exp(- n^d P(X₀ > v_n , M_An ≤ v_n)) + o(1), for some increasing sequence of sets A_n of size o(n^d). For a class of fields satisfying a local mixing condition, including m-dependent ones, the theorem holds with a constant finite A replacing A_n. The above results lead to new formulas for the extremal index for random fields.

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