The result of primary concern here is the construction of a
Hilbert space for stochastic processes. The space consisting of
random variables XX having a finite
mean-square value is (almost) a Hilbert space with inner product
EXY
X
Y
. Consequently, the distance between two random
variables XX and
YY is
dXY=EX-Y21/2
d
X
Y
X
Y
2
1/2
Now
dXY=0⇒EX-Y2=0
d
X
Y
0
X
Y
2
0
. However, this does not imply that
X=Y
X
Y
. Those sets with probability zero appear again.
Consequently, we do not have a Hilbert space unless we agree
X=Y
X
Y
means
PrX=Y=1
X
Y
1
.
Let
Xt
X
t
be a process with
EX2t<∞
X
t
2
. For each tt,
Xt
X
t
is an element of the Hilbert space just defined.
Parametrically,
Xt
X
t
is therefore regarded as a "curve" in a Hilbert space.
This curve is continuous if
limt→uEXt-Xu2=0
t
u
X
t
X
u
2
0
Processes satisfying this condition are said to be
continuous in the quadratic mean. The vector space
of greatest importance is analogous to
L
2
T
i
T
f
L
2
T
i
T
f
. Consider the collection of real-valued stochastic
processes
Xt
X
t
for which
∫
T
i
T
f
EX2tdt<∞
t
T
i
T
f
X
t
2
Stochastic processes in this collection are easily verified to
constitute a linear vector space. Define an inner product for
this space as:
E<Xt,Yt>=E∫
T
i
T
f
XtYtdt
X
t
Y
t
t
T
i
T
f
X
t
Y
t
While this equation is a valid inner product, the left-hand side
will be used to denote the inner product instead of the notation
previously defined. We take
<Xt,Yt>
X
t
Y
t
to be the time-domain inner product as
shown here. In
this way, the deterministic portion of the inner product and the
expected value portion are explicitly indicated. This
convention allows certain theoretical manipulations to be
performed more easily.
One of the more interesting results of the theory of stochastic
processes is that the normed vector space for processes
previously defined is separable. Consequently, there exists a
complete (and, by assumption, orthonormal) set
φ
i
t
φ
i
t
,
i=1…
i
1
…
of deterministic (nonrandom) functions which
constitutes a basis. A process in the space of stochastic
processes can be represented as
∀t,
T
i
≤t≤
T
f
:Xt=∑i=1∞
X
i
φ
i
t
t
T
i
t
T
f
X
t
i
1
X
i
φ
i
t
(1)
where
X
i
X
i
, the representation of
Xt
X
t
, is a sequence of random variables given by
X
i
=<Xt,
φ
i
t>
X
i
X
t
φ
i
t
or
X
i
=∫
T
i
T
f
Xt
φ
i
tdt
X
i
t
T
i
T
f
X
t
φ
i
t
Strict equality between a process and its representation cannot
be assured. Not only does the analogous issue in
L
2
0T
L
2
0
T
occur with respect to representing individual sample
functions, but also sample functions assigned a zero probability
of occurrence can be troublesome. In fact, the ensemble of any
stochastic process can be augmented by a set of sample functions
that are not well-behaved (
e.g., a sequence
of pulses) but have probability zero. In a practical sense,
this augmentation is trivial: such members of the process cannot
occur. Therefore, one says that two processes
Xt
X
t
and
Yt
Y
t
are equal almost everywhere if the distance between
∥Xt-Yt∥
X
t
Y
t
is zero. The implication is that any lack of strict
equality between the processes (strict equality means the
processes match on a sample-function-by-sample-function basis)
is "trivial."
