Analytic Properties of Random Processes
Properties :: Continuity :: Differentiation :: Integration
Differentiation
Recall that
is
differentiable
at
if
exists.
Similarly, a r.p.
is meansquare differentiable at
if
exists in the meansquare sense, that is,
If is meansquare differentiable at every , then defines another random process on the underlying probability space .
Suppose is a secondorder random process, and (m.s.). Then:
 ;

and if
then
Properties of the derivative
Suppose is meansquare differentiable with meansquare derivative . Suppose that is second order. Then:
 is also second order. (This follows from the first fact above.)

exists, and equals
.

exists and is equal to
for all
.
 exists and is equal to .

Suppose
and
is also W.S.S. Then
is
also W.S.S. Also,
and
are jointly W.S.S.
Also,
 .
 .
On the existence of the meansquare derivative
It can be shown that a
sufficient
condition for the existence of
is the existence of
at
. A
necessary
condition is the existence and
equality of the mixed partials
If is W.S.S., then these two conditions are the same. So exists if and only if
By a result we will show later, we will find that
So the existence of for a WSS process is equivalent to the condition that the second moment of the PSD of is finite.