RANDOM FIELDS AND THEIR GEOMETRY

26 1. Random fields
The corresponding variances have a nice interpretation in terms of spectral
moments when f is stationary. For example, if f has mean square
partial derivatives of orders α + β and γ + δ for α, β, γ, δ ∈ {0, 1, 2, . . . },
then 28
(1.4.34)
� α+β ∂ f(t)
E
∂αti∂ βtj · ∂γ+δf(t) ∂γtk∂ δ �
tl
= (−1) α+β ∂α+β+γ+δ ∂αti∂ βtj∂ γtk∂ δ �
�
C(t) �
tl t=0
= (−1) α+β i α+β+γ+δ
�
λ α i λ β
j λγ
kλδl ν(dλ).
Here are some important special cases of the above, for which we adopt the
shorthand fj = ∂f/∂tj and fij = ∂ 2 f/∂ti∂tj along with a corresponding
shorthand for the partial derivatives of C.
(i) fj has covariance function −Cjj and thus variance λ2ej = −Cjj(0),
where ej, as usual, is the vector with a 1 in the j-th position and zero
elsewhere.
(ii) In view of (1.4.30), and taking β = γ = δ = 0, α = 1 in (1.4.34)
(1.4.35)
f(t) and fj(t) are uncorrelated,
for all j and all t. If f is Gaussian, this is equivalent to independence.
Note that (1.4.35) does not imply that f and fj are uncorrelated as
processes. In general, for s �= t, we will have that E{f(s)fj)(t)} =
−Cj(s − t) �= 0.
(iii) Taking α = γ = δ = 1, β = 0 in (1.4.34) gives that
(1.4.36)
for all i, j, k and all t.
fi(t) and fjk(t) are uncorrelated
This concludes our initial discussion of stationarity for random fields on
R N . In the following Section we investigate what happens under slightly
weaker conditions and after that what happens when the additional assumption
of isotropy is added. In Section 1.4.7 we shall see how all of this
is really part of a far more general theory for fields defined on groups.
In Section 2.5 we shall see that, at least for Gaussian fields restricted to
bounded regions, that spectral theory is closely related to a far wider theory
of orthogonal expansions that works for general – i.e. not necessarily
Euclidean – parameter spaces
28 If you decide to check this for yourself using (1.4.19) and (1.4.20) – which is a
worthwhile exercise – make certain that you recall the fact that the covariance function
is defined as E{f(s)f(t)}, or you will make the same mistake RJA did in [1] and forget
the factor of (−1) α+β in the first line. Also, note that although (1.4.34) seems to have
some asymmetries in the powers, these disappear due to the fact that all odd ordered
spectral moments, like all odd ordered derivatives of C, are identically zero.
R N