Page 231 - Probability, Random Variables and Random Processes

CHAP. 61 ANALYSIS AND PROCESSING OF RANDOM PROCESSES

0 s t
Fig. 6-2 Shifted unit step function.

Note that although a m.s. derivative does not exist for the Wiener process, we can define a generalized
derivative of the Wiener process (see Prob. 6.20).

6.10. Show that if X(t) is a normal random process for which the m.s. derivative Xt(t) exists, then Xf(t)
is also a normal random process.
Let X(t) be a normal random process. Now consider

Then, n r.v.'s Y,(t,), I&), . . . , Y,(t,,) are given by a linear transformation of the jointly normal r.v.'s X(t,),
X(t, + E), X(t2), X(t2 + E), . . . , X(t,), X(tn + E). It then follows by the result of Prob. 5.60 that Y,(t,), Y,(t,),
. . . , Y,(tn) are jointly normal r.v.'s, and hence Y,(t) is a normal random process. Thus, we conclude that the
m.s. derivative X'(t), which is the limit of Y,(t) as E + 0, is also a normal random process, since m.s. con-
vergence implies convergence in probability (see Prob. 6.5).

6.11. Show that the m.s. integral of a random process X(t) exists if the following integral exists:

A m.s. integral of X(t) is defined by [Eq. (641

Again using the Cauchy criterion, the m.s. integral Y(t) of X(t) exists if

As in the case of the m.s. derivative [Eq. (6.1 1 I)], expanding the square, we obtain

{[ k
E 1 X(ti) Ati - C X(tk) At, 1'1
i

and Eq. (6.1 20') holds if
lim 1 R,(ti, t,) Ati Atk
Ati, Atr+ 0 i k