On Extreme Values in Stationary Random Fields

Summary

This paper develops distributional extremal theory for maxima M _T = max(X _t: 0 ⩽ t ⩽ T) of a stationary random field X _t. A general form of “extremal types theorem” is proven and shown to apply to M _T under very weak dependence restrictions. That is, any non-degenerate distributional limit for the normalized family a _T(M_T - b _T) (a _T > 0) must be one of the three classical types. Domain of attraction criteria are discussed.

The dependence structure used here for fields involves a potentially very weak type of strong-mixing, “Coordinatewise (Cw) mixing”) using mild individual “past-future” conditions in each coordinate direction. Together with careful control of numbers and sizes of sets involved, this avoids the over-restrictive nature of common generalizations of mixing conditions to apply to random fields. Futher, the conditions may be readily adpated to deal with other quite general problems of Centeral Limit type (cf. [6]).