RANDOM FIELDS AND THEIR GEOMETRY

34 1. Random fields
where the Wml are uncorrelated with variance depending only on m. This,
of course, is simply (1.4.43) once again, derived from a more general setting.
Similarly, the covariance function can be written as
(1.4.47)
C(θ1, θ2) = C(θ12) =
∞�
m=0
σ 2 mC (N−1)/2
m (cos(θ12)),
where θ12 is the angular distance between θ1 and θ2, and the C N m are the
Gegenbauer polynomials.
Other examples for the LCA situation follow in a similar fashion from
(1.4.44) and (1.4.45) by knowing the structure of the dual group Γ.
The general situation is much harder and, as has already been noted,
relies heavily on knowing the representation of T . In essence, given a representation
of G on a GL(H) for some Hilbert space H, one constructs a
(left or right) stationary random field on G via the canonical white noise
on H. The construction of Lemma 1.4.2 with T = G and H = L 2 (G, µ) can
be thought of as a special example of this approach. For further details,
you should go to the references we gave at the beginning of this Section.
1.5 Non-Gaussian fields
This section should be of interest to most readers and is crucial for those
who care about applications,
Up until now, we have concentrated very heavily on Gaussian random
fields. The one point where we departed somewhat from this theme was
in the discussion on stationarity, where normality played a very limited
rôle. In the following Chapter we shall concentrate exclusively on Gaussian
fields.
Despite, and perhaps because of, this it is time to take a moment to
explain both the centrality of Gaussian fields and how to best move away
from them.
It will become clear as you progress through this book that while appeals
to the Central Limit Theorem may be a nice way to justify concentrating on
the Gaussian case, the real reason for this concentration is somewhat more
mundane. The relatively uncomplicated form of the multivariate Gaussian
density (and hence finite-dimensional distributions of Gaussian fields)
makes it a reasonably straightforward task to carry out detailed computations
and allows one to obtain explicit results and precise formulae for many
facets of Gaussian fields. It is difficult to over-emphasise the importance of
explicit results for applications. There is a widespread belief among modern
pure mathematicians that the major contribution they have to make
to ‘Science’ is the development of ‘Understanding’, generally at the expense
of explicit results. Strangely enough, most subject matter scientists
do not share the mathematicians’ enthusiasm for insight. They generally