RANDOM FIELDS AND THEIR GEOMETRY

28 1. Random fields
Proof. Isotropy implies that C can be written as a function on R+ only.
Let τ be any positive real. We shall show that C(τ) ≥ −C(0)/N.
Choose any t1, . . . , tN+1 in R N for which |ti − tj| = τ for all i �= j. Then,
by (1.4.38),
⎧�
�
⎨�N+1
2
�
� �
�
E � X(tk) �
⎩�
�
⎫ ⎬
= (N + 1)[C(0) + C(τ)].
⎭
k=1
Since this must be positive, the result follows. ✷
The restriction of isotropy also has significant simplifying consequences
for the spectral measure ν of (1.4.16). Let θ : RN → RN be a rotation, so
that |θ(t)| = |t| for all t. Isotropy then implies C(t) = C(θ(t)) and so the
Spectral Distribution Theorem implies
�
RN e i〈t,λ〉 �
ν(dλ) =
RN e i〈θ(t),λ〉 (1.4.39)
ν(dλ)
�
=
RN e i〈t,θ(λ)〉 ν(dλ)
�
= e i〈t,λ〉 νθ(dλ),
where νθ is the push-forward of ν by θ defined by νθ(A) = ν(θ −1 A). Since
the above holds for all t it follows that ν ≡ νθ; i.e. ν, like C, is invariant
under rotation. Furthermore, if ν is absolutely continuous, then its density
is also dependent only on the modulus of its argument.
An interesting consequence of this symmetry is that an isotropic field
cannot have all the probability of its spectral measure concentrated in one
small region in R N away from the origin. In particular, it is not possible
to have a spectral measure degenerate at one point, unless that point is
the origin. The closest the spectral measure of an isotropic field can come
to this sort of behaviour is to have all its probability concentrated in an
annulus of the form
{λ ∈ R N : a ≤ |λ| ≤ b}.
In such a case it is clear from (1.4.26) and (1.4.27) that the field itself
is then composed of a ‘sum’ of waves travelling in all directions but with
wavelengths between 2π/b and 2π/a only.
Another consequence of isotropy is the the spherical symmetry of the
spectral measure significantly simplifies the structure of the spectral moments
and so the correlations between various derivatives of f. In particular,
it follows immediately from (1.4.34) that
(1.4.40)
R N
E {fi(t)fj(t)} = −E {f(t)fij(t)} = λ2δij