RANDOM FIELDS AND THEIR GEOMETRY

12 1. Random fields
to meet our specific needs and that both style and content are occasionally
a little eclectic. In other words, you should go elsewhere for fuller, more
standard treatments. References will be given along the way.
Although our primary interest lies in the study of real valued random
fields it is mathematically more convenient to discuss stationarity in the
framework of complex valued processes. Hence, unless otherwise stated,
we shall assume throughout this section that f(t) = (fR(t) + ifI(t)) takes
values in the complex plane C and that E{�f(t)� 2 } = E{f 2 R (t)+f 2 I
(t)} < ∞.
(Both fR and fI are, obviously, to be real valued.) As for a definition
of normality in the complex scenario, we first define a complex random
variable to be Gaussian if the vector of its two components is bivariate
Gaussian13 . A complex process f is Gaussian if �
i αtif ′ ti is a complex
Gaussian variable for all sequences {ti} and complex {αti }.
We also need some additional structure on the parameter space T . In
particular, we need that it have a group structure14 and an operation with
respect to which the field is stationary. Consequently, we now assume that
T has such a structure, ‘+’ represents the binary operation on T and ‘−’
repesents inversion. As usual, t − s = t + (−s). For the moment, we need
no further assumptions on the group.
Since ft ∈ C, it follows that the mean function m(t) = E{f(t)} is also
complex valued, as is the covariance function, which we redefine for the
complex case as
(1.4.1)
C(s, t) ∆ �
�
= E [f(s) − m(s)] [f(t) − m(t)] ,
with the bar denoting complex conjugation.
Some basic properties of covariance functions follow immediately from
(1.4.1):
• C(s, t) = C(t, s), which becomes the simple symmetry C(s, t) =
C(t, s) if f (and so C) is real valued.
• For any k ≥ 1, t1, . . . , tk ∈ T and z1, . . . , zk ∈ C, the Hermitian
form �k �k i=1 j=1 C(ti, tj)zizj is always real and non-negative. We
summarise this, as before, by saying that C is non-negative definite.
(The second property follows from the equivalence of the double sum to
E{� �k i=1 [f(ti) − m(ti)]zi�2 }.)
Suppose for the moment that T is Abelian. A random field f is called
strictly homogeneous or strictly stationary over T , with respect to the group
13It therefore folows that a complex Gaussian variable X = XR + iXI is defined by
five parameters: E{XI}, E{XR}, E{X2 I }, E{X2 R } and E{XIXR}.
14If stationarity is a new concept for you, you will do well by reading this Section the
first time taking T = RN with the usual notions of addition and subtraction.