Page 227 - Schaum's Outlines - Probability, Random Variables And Random Processes

220 ANALYSIS AND PROCESSING OF RANDOM PROCESSES [CHAP 6

6.2. Show that a WSS random process X(t) is m.s. continuous if and only if its autocorrelation
function R,(z) is continous at z = 0.
If X(t) is WSS, then Eq. (6.1 07) becomes

Thus if RX(r) is continuous at z = 0, that is,
lirn [RX(&) - RX(0)] = 0
&+O
then lirn E([X(t + E) - X(t)I2) = 0
&+O
that is, X(t) is m.s. continuous. Similarly, we can show that if X(t) is m.s. continuous, then by Eq. (6.108),
RAT) is continuous at z = 0.

6.3. Show that if X(t) is m.s. continuous, then its mean is continuous; that is,
lirn px(t + E) = pX(t)
c-+o
We have
Var[X(t + E) - X(t)] = E([X(t + E) - X(t)I2} - (E[X(t + E) - ~(t)])' 2 0
Thus E([X(t + E) - X(t)I2) 2 (E[X(t + E) - X(t)])2 = [px(t + E) - px(t)12
If X(t) is m.s. continuous, then as E -+ 0, the left-hand side of the above expression approaches zero. Thus
lirn [px(t + E) - px(t)] = 0 or lirn [p,(t + E) = pdt)
c-0 E+O

6.4. Show that the Wiener process X(t) is m.s. continuous.

From Eq. (5.64), the autocorrelation function of the Wiener process X(t) is given by

Thus, we have

Since lim max(cl , c2) = 0
El. 62-0
RAt, s) is continuous. Hence the Wiener process X(t) is m.s. continuous.

6.5. Show that every m.s. continuous random process is continuous in probability.
A random process X(t) is continuous in probability if, for every t and a > 0 (see Prob. 5.62),
lirn P{ I X(t + E) - X(t) I > a) = 0
e+O
Applying Chebyshev inequality (2.97) (Prob. 2.37), we have

Now, if X(t) is m.s. continuous, then the right-hand side goes to 0 as E -* 0, which implies that the left-hand
side must also go to 0 as E -+ 0. Thus, we have proved that if X(t) is m.s. continuous, then it is also
continuous in probability.

6.6. Show that a random process X(t) has a m.s. derivative X'(t) if a2~,(t, s)/at as exists at s = t.