RANDOM FIELDS AND THEIR GEOMETRY

32 1. Random fields
where C is now a function from R2 to C. In such a situation the methods
of the previous proof suffice to show that C can be written in the form
where
C(t, x) =
GN (x) =
� ∞ � ∞
−∞
0
e itν GN (λx) µ(dν, dλ),
� � (N−2)/2 � �
2
N
Γ J (N−2)/2(x)
x
2
and µ is a measure on the half-plane R+ × R N .
By now it should be starting to become evident that all of these representations
must be special cases of some general theory, that might also
be able to cover non-Euclidean parameter spaces. This is indeed the case,
although for reasons that will soon be explained the general theory is such
that, ultimately, each special case requires almost individual treatment.
1.4.7 Stationarity over groups
We have already seen in Section 1.4.2 that the appropriate setting for stationarity
is when the parameter set has a group structure. In this case it
made sense, in general, to talk about left and right stationarity (cf. (1.4.13)
and (1.4.14)). Simple ‘stationarity’ requires both of these and so makes
most sense if the group is Abelian (commutative).
In essence, the spectral representation of a random field over a group
is intimately related to the representation theory of the group. This, of
course, is far from being a simple subject. Furthermore, its level of difficulty
depends very much on the group in question and so it is correspondingly
not easy to give a general spectral theory for random fields over groups.
The most general results in this area are in the paper by Yaglom [110]
already mentioned above and the remainder of this Subsection is taken
from there 33 .
We shall make life simpler by assuming for the rest of this Subsection
that T is a locally compact, Abelian (LCA) group. As before, we shall denote
the binary group operation by + while − denotes inversion. The Fourier
analysis of LCA groups is well developed (e.g. [80]) and based on characters.
A homomorphism γ from T to the multiplicative group of complex numbers
is called a character if �γ(t)� = 1 for all t ∈ T and if
γ(t + s) = γ(t) γ(s), s, t ∈ T.
33 There is also a very readable, albeit less exhaustive, treatment in Hannan [43].
In addition, Letac [60] has an elegant exposition based on Gelfand pairs and Banach
algebras for processes indexed by unimodular groups, which, in a certain sense, give a
generalisation of isotropic fields over R N . Ylinen [111] has a theory for noncommutative
locally compact groups that extends the results in [110].