On maxima of stationary fields

Abstract

Let $\{X_{\mathbf{n}}:\mathbf{n}\in {\mathbb{Z}}^d\}$ be a weakly dependent stationary field with maxima $M_A:=\sup \{X_{\mathbf{i}}:\mathbf{i}\in A\}$ for finite $A\subset {\mathbb{Z}}^d$ and $M_{\mathbf{n}}:=\sup \{X_{\mathbf{i}}:1\le \mathbf{i}\le \mathbf{n}\}$ for $\mathbf{n}\in {\mathbb{N}}^d$. In a general setting we prove that $P(M_{(n,n,\dots ,n)}\le v_n)=\exp (-n^{d}P(X_0>v_n,M_{A_n}\le v_n))+o(1)$, 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.

Accepted for publication by the Applied Probability Trust in Journal of Applied Probability 56.4