EXTREMES OF MOVING AVERAGES AND MOVING MAXIMA ON A
REGULAR LATTICE
Abstract: We study the extremal behaviour of spatial moving averages and moving maxima on a
regular discrete grid. Our main assumption is that these random fields are stationary and
regularly varying with the tail index . Using the asymptotic theory for point
processes we characterise the limiting behaviour of their extremes over an increasing
grid. Our approach builds on the results of Davis and Resnick concerning linear
processes.
By analogy to the analysis of time series data, an appropriate Hill estimator of the tail
index can be defined. We exhibit a sufficient condition for the consistency of this estimator in
a certain class of spatial lattice models. Finally, we show that this condition holds for the
models in our title.
2000 AMS Mathematics Subject Classification: Primary: 60G70; Secondary:
60G55.
Keywords and phrases: Regular variation, point processes, stationary random fields,
regular grid, extreme value theory.