RANDOM FIELDS AND THEIR GEOMETRY

4 1. Random fields
with mj = E{Xj} and a non-negative definite 4 d × d covariance matrix
C, with elements cij = E{(Xi − mi)(Xj − mj)}, such that the probability
density of X is given by
(1.2.3)
ϕ(x) =
1
(2π) d/2 1
e− 2
|C| 1/2 (x−m)C−1 (x−m) ′
,
where |C| = det C is the determinant 5 of C. Consistently with the onedimensional
case, we write this as X ∼ N(m, C), or X ∼ Nd(m, C) if we
need to emphasise the dimension.
In view of (1.2.3) we have that Gaussian distributions are completely
determined by their first and second order moments and that uncorrelated
Gaussian variables are independent. Both of these facts will be of crucial
importance later on.
While the definitions are fresh, note for later use that it is relatively
straightforward to check from (1.2.3) that the characteristic function of a
multivariate Gaussian X is given by
(1.2.4)
φ(θ) = E{e i〈θ,X′ 〉 } = e i〈θ,m ′ 〉− 1
2 θCθ′
.
where θ ∈ R d .
One consequence of the simple structure of φ is the fact that if {Xn}n≥1
is an L 2 convergent 6 sequence of Gaussian vectors, then the limit X must
also be Gaussian. Furthermore, if Xn ∼ N(mn, Cn), then
(1.2.5)
|mn − m| 2 → 0, and �Cn − C� 2 → 0,
as n → ∞, where m and C are the mean and covariance matrix of the
limiting Gaussian. The norm on vectors is Euclidean and that on matrices
any of the usual. The proofs involve only (1.2.4) and the continuity theorem
for convergence of random variables.
One immediate consequence of either (1.2.3) or (1.2.4) is that if A is any
d × d matrix and X ∼ Nd(m, C), then
(1.2.6)
AX ∼ N(mA, A ′ CA).
4 A d×d matrix C is called non-negative definite (or positive semi-definite) if xCx ′ ≥ 0
for all x ∈ R d . A function C : T × T → R is called non-negative definite if the matrices
(C(ti, tj)) n i,j=1 are non-negative definite for all 1 ≤ n < ∞ and all (t1, . . . , tn) ∈ T n .
5 Just in case you have forgotten what was in the Preface, here is a one-time reminder:
The notation | | denotes any of ‘absolute value’, ‘Euclidean norm’, ‘determinant’ or
‘Lebesgue measure’, depending on the argument, in a natural fashion. The notation � �
is used only for either the norm of complex numbers or for special norms, when it usually
appears with a subscript.
6 That is, there exists a random vector X such that E{|Xn − X| 2 } → 0 as n → ∞.