Usually the above concept of whiteness is sufficient, but a
much stronger definition is as follows:

Pick a set of times
t1
t2
…
tN
t1
t2
…
tN
to sample
Xt
X
t
.

If, for *any choice* of
t1
t2
…
tN
t1
t2
…
tN
with NN finite, the
random variables
X
t1
X
t1
,
X
t2
X
t2
, ……
X
tN
X
tN
are jointly independent, i.e. their
joint pdf is given by

f
X
(
t1
)
,
X
(
t2
)
,
…
X
(
tN
)
x1
x2
…
xN
=∏i=1N
f
X
(
ti
)
xi
f
X
(
t1
)
,
X
(
t2
)
,
…
X
(
tN
)
x1
x2
…
xN
i
1
N
f
X
(
ti
)
xi

(4)
and the marginal pdfs are identical, i.e.

f
X
(
t1
)
=f
X
(
t2
)
=…=f
X
(
tN
)
=
fX
f
X
(
t1
)
f
X
(
t2
)
…
f
X
(
tN
)
fX

(5)
then the process is termed

Independent and Identically
Distributed (i.i.d).

If, in addition,
fX
fX
is a pdf with zero mean, we have a Strictly
White Noise Process.

An i.i.d. process is 'white' because the variables
X
ti
X
ti
and
X
tj
X
tj
are jointly independent, even when separated by an
infinitesimally small interval between
ti
ti
and
tj
tj
.