On the Maximum and Minimum of a Stationary Random Field

Abstract

We determine the class of nondegenerate joint-limiting distributions for the maximum and minimum of stationary random fields $\mathbf{X} = \left \{X_{\mathbf{n}}\right \}_{\mathbf{n\in {\mathbb{N}}^{2}}}$ satisfying a long-range dependence restriction for each coordinate direction at a time. Unlike the classical result for i.i.d. random fields the maximum and minimum of X may be asymptotically dependent. We also give a sufficient condition for the asymptotic independence of the maximum and minimum. Additional conditions are given in order to obtain the asymptotic independence of the locations of maximum and minimum.