Page 91 - Special Topic Session (STS) - Volume 3

STS517 Zaoli C. et al.

Extreme value theory for long range dependent

stable random fields
Zaoli Chen, Gennady Samorodnitsky
Cornell University, USA

Abstract
We study the extremes for a class of a symmetric stable random fields with
long range dependence.

Keywords
Random field; extremal limit theorem; random sup measure; random closed
set; long range dependence; stable law; heavy tails.

1. Introduction
Extreme value theorems describe the limiting behaviour of the largest
values in increasingly large collections of random variables. The classical
extremal theorems, beginning with Fisher and Tippett (1928) and Gnedenko
(1943), deal with the extremes of i.i.d. (independent and identically distributed)
random variables. The modern extreme value theory techniques allow us to
study the extremes of dependent sequences; see Leadbetter et al. (1983) and
the expositions in Coles (2001) and de Haan and Ferreira (2006). The effect of
dependence on extreme values can be restricted to a loss in the effective
sample size, through the extremal index of the sequence. When the
dependence is sufficiently long, more significant changes in extreme value
may occur; see e.g. Samorodnitsky (2004), Owada and Samorodnitsky (2015b).
The present paper aims to contribute to our understanding of the effect of
memory on extremes when the time is of dimension larger than 1, i.e. for
random fields.
We consider a discrete time stationary random field = ( , ∈ ℤ ). For

= ( , . . . , ) ∈ ℕ we would like to study the extremes of the random field

1
over growing hypercubes of the type

,

1991 Mathematics Subject Classification. Primary 60G60, 60G70, 60G52.
Key words and phrases. Random field, extremal limit theorem, random sup measure, random closed
set, long range dependence, stable law, heavy tails.
This research was partially supported by the NSF grant DMS-1506783 and the ARO grant W911NF-18 -
10318 at Cornell University.

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