Page 95 - Special Topic Session (STS) - Volume 3

STS517 Zaoli C. et al.
Even though we are using the same notation T for operators acting on
different spaces, the meaning will always be clear from the context. Note that
each individual left shift on (ℤ , ℬ(ℤ ), ) is measure preserving (because
ℕ
ℕ

()
each ( ) is an invariant measure.) It is also conservative and ergodic by
∈ℤ

Theorem 4.5.3 in Aaronson (1997). Therefore, the group action = { : ∈
ℤ } is conservative, ergodic and measure preserving on (, ℇ, ).

Equipped with a measure preserving group action on the space (, ℰ) we
can now define a stationary symmetric α-stable random field by
(2.5) = ∫ ο ()(), ∈ ℤ ,

where is a random measure on(, ℰ) with control measure µ, and
(2.6) () = 1( () ∈ , = 1, … , ), = ( (1) , … , () ).

where = { ∈ ℤ ∶ = 0}. Clearly, ∈ (), which guarantees that the
ℤ
∝
0
integral in (2.5) is well defined. We refer the reader to Samorodnitsky and
Taqqu (1994) for general information on stable processes and integrals with
respect to stable measures, and to Rosiński (2000) on more details on
stationary stable random fields and their representations.
The random field model defined by (2.5) is attractive because the key
parameters involve in its definition have a clear intuitive meaning: the index of
stability 0 < α < 2 is responsible for the heaviness of the tails, while 0 < <

1, = 1, … , (defined in (2.2)) are responsible for the “length of the memory”.
The latter claim is not immediately obvious, but its (informal) validity will
become clearer in the sequel.
The following array of positive numbers will play the crucial role in the
extremal limit theorems in this paper. Denote for = 1, 2, …. and = 1, … ,

() 1/
= ( ({x: = 0 for some = 0, 1, … , n})) ,

and let

Then = ( ), where

= {x = (x (1) , … , x (d) ) ∈ : () = 0 for some 0 ≤ ≤ , each = 1, … , }.

Therefore, we can define, for each ∈ ℕ , a probability measures on (, ℇ)

n
by
(2.8) (∙) = n − (∙ ∩ _ ).
n
This probability measure allows us to represent the restriction of the stationary
SαS random field in (2.5) to the hypercube [, ] = { ≤ ≤ } as a series,
described below, and that we will find useful in the sequel. It is useful to note
also that the measure is the product measure of probability measures

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