Extreme Value Theory for Long-Range-Dependent Stable Random Fields
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Extreme Value Theory for Long-Range-Dependent Stable Random Fields Zaoli Chen1 · Gennady Samorodnitsky2 Received: 14 October 2018 / Revised: 1 October 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We study the extremes for a class of a symmetric stable random fields with long-range dependence. We prove functional extremal theorems both in the space of sup measures and in the space of càdlàg functions of several variables. The limits in both types of theorems are of a new kind, and only in a certain range of parameters, these limits have the Fréchet distribution. Keywords Random field · Extremal limit theorem · Random sup measure · Random closed set · Long-range dependence · Stable law · Heavy tails Mathematics Subject Classification (2010) Primary 60G60 · 60G70 · 60G52
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 [6] and Gnedenko [8], 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. [10] and the expositions in Coles [4] and Haan and Ferreira [5]. The effect of
This research was partially supported by the NSF Grant DMS-1506783 and the ARO Grant W911NF-18-10318 at Cornell University.
B
Gennady Samorodnitsky [email protected] Zaoli Chen [email protected]
1
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
2
School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA
123
Journal of Theoretical Probability
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 [24] and Owada and Samorodnitsky [18]. 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 X = X t , t ∈ Zd . For n = (n 1 , . . . , n d ) ∈ Nd , we would like to study the extremes of the random field over growing hypercubes of the type [0, n] = 0 ≤ k ≤ n , n → ∞ , where 0 is the vector with zero coordinates, the notation s ≤ t for vectors s = (s1 , . . . , sd ) and t = (t1 , . . . , td ) means that si ≤ ti for all i = 1, . . . , d, and the notation n → ∞ means that all d components of the vector n tend to infinity. Denote Mn = max X t . 0≤k≤n
What limit theorems does the array (Mn ) satisfy? It was shown by Leadbetter and Rootzén [11] that under appropriate strong mixing conditions, only the classical three types of limiting distributions (Gumbel, Fréchet and Weibull) may appear (even when forcing n → ∞ along a monotone curve). In the case, when the marginal distributio
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