| Extreme value theory for random fields is developed along with Central Limit Theory for random functionals defined above high levels, e.g. exceedance measures and related tail estimators. So far many research results for Extreme Values have been developed concerning random processes, but, few for random fields. The Asymptotic Theories for measures of exceedance by random fields are Conveniently developed here within the general framework of random additive functions. One of the challenges of dealing with random fields is in the provision of appropriate long range dependence conditions. We have recently shown that potentially more tractable and weaker conditions may be obtained by the replacement of one "global" dependence restriction by one in each coordinate direction for which past-future separation can be utilized.;A general version of the classical "Extremal Types Theorem" is given for maxima of stationary random fields and the existence of the so-called Extremal Index is shown under new forms of long range dependence condition. In particular the extremal properties of Gaussian fields under Berman's well-known covariance condition are shown to hold unchanged under this weaker dependence condition. A proof of the Type I limit for maxima of Gaussian processes is given, utilizing and amplifying recent methods of Albin for the tail distribution of the maximum in a fixed interval. Finally, we present simulation results for the constant Halpha that appears in the limit of the tail probability of maxima of normal process, but whose exact value is known only for alpha = 1, 2. |
