RANDOM FIELDS AND THEIR GEOMETRY

20 1. Random fields
Theorem 1.4.3 (Spectral distribution theorem) A continuous function
C : RN → C is non-negative definite (i.e. a covariance function) if
and only if there exists a finite measure ν on BN such that
�
C(t) = e i〈t,λ〉 (1.4.16)
ν(dλ),
for all t ∈ R N .
R N
With randomness in mind, we write σ 2 = C(0) = ν(R N ). The measure ν
is called the spectral measure (for C) and the function F : R N → [0, σ 2 ]
given by
F (λ) ∆ �
N�
�
= ν (−∞, λi] , λ = (λ1, . . . , λN ) ∈ R N ,
i=1
is called the spectral distribution function 20 . When F is absolutely continuous
the corresponding density is called the spectral density.
The spectral distribution theorem is a purely analytic result and would
have nothing to do with random fields were it not for the fact that covariance
functions are non-negative definite. Understanding of the result
comes from the spectral representation theorem (Theorem 1.4.4) for which
we need some preliminaries.
Let ν be a measure on R N and WR and WI be two independent (ν/ √ 2)noises,
so that (1.4.4)–(1.4.6) hold for both WR and WI with ν/ √ 2 rather
than ν. (The factor of 1/ √ 2 is so that (1.4.18) below does not need a
unaesthetic factor of 2 on the right hand side.). There is no need at the
moment to assume that these noises are Gaussian. Define a new, C-valued
noise W by writing
W (A) ∆ = WR(A) + iWI(A),
for all A ∈ B N . Since E{W (A)W (B)} = ν(A ∩ B), we call W a complex
ν-noise. For f : R N → C with �f� ∈ L 2 (ν) we can now define the complex
integral W (f) by writing
(1.4.17)
W (f) ≡
≡
�
�
R N
R N
f(λ) W (dλ)
(fR(λ) + ifI(λ)) (WR(dλ) + iWI(dλ))
∆
= [WR(fR) − WI(fI)] + i [WI(fR) + WR(fI)]
20 Of course, unless ν is a probability measure, so that σ 2 = 1, F is not a distribution
function in the usual usage of the term.