On maxima of stationary fields

TL;DR

This paper investigates the asymptotic behavior of maxima in stationary random fields, deriving formulas for their distribution and extremal index under weak dependence and mixing conditions.

Contribution

It provides a general asymptotic formula for the distribution of maxima in stationary fields, extending to fields with local mixing conditions and offering new extremal index formulas.

Findings

Asymptotic distribution of maxima expressed via tail probabilities.

Extension to fields with local mixing conditions.

New formulas for the extremal index in random 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 $M_{\mathbf{n}} := \sup\{X_{\mathbf{i}} : \mathbf{1} \leq \mathbf{i} \leq \mathbf{n} \}$ for $\mathbf{n}\in\mathbb{N}^d$. In a general setting we prove that $P(M_{(n,n,\ldots, n)} \leq v_n) = \exp(- n^d P(X_{\mathbf{0}} > v_n , M_{A_n} \leq v_n)) + o(1)$, 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.