RANDOM FIELDS AND THEIR GEOMETRY

16 1. Random fields
1.4.2 Moving averages
We now return to the main setting of this Section, in which T is an Abelian
group under the binary operation + and − represents inversion. Let ν be
a Haar measure on (T, T ) (assumed to be σ-finite) and take F : T → R in
L2 (ν). If W is a ν-noise on T , then the random process
f(t) ∆ �
(1.4.12)
= F (t − s) W (ds),
T
is called a moving average of W , and we have the following simple result:
Lemma 1.4.1 Under the preceeding conditions, f is a stationary random
field on T . Furthermore, if W is a Gaussian noise, then f is also Gaussian.
Proof. To establish stationarity we must prove that
E {f(t)f(s)} = C(t − s)
for some C. However, from (1.4.11) and the invariance of ν under the group
operation,
�
E {f(t)f(s)} = F (t − u)F (s − u) ν(du)
T �
= F (t − s + v)F (v) ν(dv)
T
∆
= C(t − s),
and we are done.
If W is Gaussian, then we have already noted when defining stochastic
integrals that f(t) = W (F (t−·)) is a real-valued Gaussian random variable
for each t. The same arguments also show that f is Gaussian as a process. ✷
A similar but slightly more sophisticated construction also yields a more
general class of examples, in which we think of the elements g of a group
G acting on the elements t of an underlying space T . This will force us to
change notation a little and, for the argument to be appreciated in full, to
assume that you also know a little about manifolds. If you do not, then you
can return to this example later, after having read Chapter 3, or simply
take the manifold to be RN . In either case, you may still want to read the
very concrete and quite simple examples at the end of this subsection now.
Thus, taking the elements g of a group G acting on the elements t of an
underlying space T , we denote the identity element of G by e and the left
and right multiplication maps by Lg and Rg. We also write Ig = Lg ◦ R−1 g
for the inner automorphism of G induced by g.
Since we are now working in more generality, we shall also drop the commutativity
assumption that has been in force so far. This necessitates some