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STS517 Zaoli C. et al.

(1)
()
ℤ

ℤ
on(ℤ , ℬ(ℤ )) : = 1 × . × for = ( , … , ) ∈ ℕ ,for where for =

1

1, … , and n ≥ 0,
−
(2.9) (∙) = ( ) (∙ ∩ {x ∈ ℤ : = 0 for some 0 ≤ ≤ }).
()
()
ℤ

The restriction of the stationary SαS random field X in (2.5) to the
hypercube [0,n] admits, in law, the series representation

d
with A = A×·×A the direct product of d copies of A and A is in (2.6), where the
constant Cα is the tail constant of the α-stable random variable:

Furthermore,{ } is a iid sequence of Rademacher random variables, {Γ } is the

sequence of the arrival times of a unit rate Poisson process on (0, ∞), and
{ } are iid -valued random elements with common law . The sequence

,
{ },{Γ } and { } are independent. See Samorodnitsky and Taqqu (1994) for
,

details.

References
1. J. Aaronson (1997): An Introduction to Infinite Ergodic Theory, volume 50
of Mathematical Surveys and Monographs. American Mathematical
Society, Providence.
2. Chakrabarty and P. Roy (2013): Group theoretic dimension of stationary
symmetric α-stable random fields. Journal of Theoretical Probability
26:240–258.
3. S. Coles (2001): An Introduction to Statistical Modeling of Extreme
Values. Springer, New York.
4. L. de Haan and A. Ferreira (2006): Extreme Value Theory: An
Introduction. Springer, New York.
5. R. A. Fisher and L. Tippett (1928): Limiting forms of the frequency
distributions of the largest or smallest member of a sample. Proceedings
of Cambridge Philisophical Society 24:180–190.
6. Gnedenko (1943): Sur la distribution limite du terme maximum d’une serie
aleatoire. Annals of Mathematics 44:423–453.
7. Lacaux and G. Samorodnitsky (2016): Time-changed extremal process as
a random sup measure. Bernoulli 22:1979–2000.

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