RANDOM FIELDS AND THEIR GEOMETRY

24 1. Random fields
FIGURE 1.4.1. The elementary wave form cos(λ1t1 + λ2t2) in R 2 .
is, the two sets of waves are out of phase by half a wavelength. As in
the one-dimensional case, the amplitudes of the components cos(〈λi, t〉)
and sin(〈λi, t〉) are given by the random variables |W1(Λi)| and |W2(Λi)|.
Figure 1.4.3 shows what a sum of 10 such components looks like, when
the λi are chosen randomly in (−π, π] 2 and the Wj(λi) are independent
N(0, 1).
FIGURE 1.4.2. A more realistic surface based on (1.4.25), along with contour
lines at the zero level.
1.4.4 Spectral moments
Since they will be very important later on, we now take a closer look at
spectral measures and, in particular, their moments. It turns out that these
contain a lot of simple, but very useful, information. Given the the spectral
representation (1.4.20); viz.
�
C(t) = e i〈t,λ〉 (1.4.28)
ν(dλ),
we define the spectral moments
�
∆
=
(1.4.29)
λi1...iN
R N
R N
λ i1
1
· · · λiN
N ν(dλ),
for all (i1, . . . , iN) with ij ≥ 0. Recalling that stationarity implies that
C(t) = C(−t) and ν(A) = ν(−A), it follows that the odd ordered spectral