RANDOM FIELDS AND THEIR GEOMETRY

36 1. Random fields
which is independent of σ 2 .
(iii) F field: Take d = n + m and
(1.5.4)
F (x) = m �n 1 x2i n �n+m n+1 x2 .
i
The corresponding random field is known as the F field with n and
m degrees of freedom and has marginal density, for x > 0, given by
ϕF (x) ∆ = nn/2 m m/2
B(n/2, m/2)
where B is the usual Beta function.
xn/2−1 ,
(m + nx) (n+m)/2
Consider the issue of stationarity for each of these three examples. From
their definitions, we could evaluate their covariance functions and check
them for weak, L 2 stationarity. However, this would be rather foolish, since
the strong stationarity of their Gaussian components clearly implies strong
stationarity for them as well. Sample path smoothness of various degrees
will also follow immediately from any assumed smoothness on the paths of
g, since the transformations F are C ∞ everywhere except at the origin 34 .
Thus it should be reasonably clear that elementary properties of f pass
over quite simply to f = F (g) as long as F is ‘nice’.
A more interesting question for us will be how to study the excursion
sets (1.1.1) of f. There are two ways to develop this theory. In the past,
the standard approach was to treat each particular F as a special case and
to handle it accordingly. This invariably involved detailed computations,
which were always related to the underlying Gaussian structure of g and to
the specific transformation F . This approach lead to a plethora of papers,
which provided not only a nice theory but also a goodly number of PhD
theses, promotions and tenure successes. In JT’s thesis [94] (see also [95])
another approach was taken, based, in essence, on the observation that
(1.5.5)
Au(f, T ) = Au(F (g), T ) = {t ∈ T : (F ◦ g)(t) ≥ u}
= {t ∈ T : g(t) ∈ F −1 [u, ∞)}.
Thus, the excursion set of a real valued non-Gaussian f = F ◦ g above a
level u is equivalent to the excursion set for a vector valued Gaussian g in
the manifold F −1 [u, ∞). We shall see in Chapter 5 how this approach can
be used to separate out probability computations from geometric computations
based on the properties of F and so obtain an elegant theory for
this non-Gaussian scenario.
34 The pole at the origin will actually cause problems in some cases, such as T = R N
and d = N in the χ 2 and T cases, as well as m = N in the F case, but we can worry
about that later.