Page 192 - Probability, Random Variables and Random Processes

RANDOM PROCESSES [CHAP 5

Hence, all finite-order distributions of a Markov process can be completely determined by the second-order
distribution.

5.26. Show that if a normal process is WSS, then it is also strict-sense stationary.
By Eq. (5.29), a normal random process X(t) is completely characterized by the specification of the
mean E[X(t)] and the covariance function Kx(t, s) of the process. Suppose that X(t) is WSS. Then, by Eqs.
(5.21) and (5.22), Eq. (5.29) becomes

Now we translate all of the time instants t,, t,, . . . , t, by the same amount z. The joint characteristic
function of the new r.v.'s X(ti + z), i = 1, 2, . . . , n, is then

which indicates that the joint characteristic function (and hence the corresponding joint pdf) is unaffected by
a shift in the. time origin. Since this result holds for any n and any set of time instants (ti E T, i = 1,2, . . . , n),
it follows that if a normal process is WSS, then it is also strict-sense stationary.

5.27. Let (X(t), - oo < t < oo} be a zero-mean, stationary, normal process with the autocorrelation
function

(0 otherwise
Let {X(ti), i = 1,2, . . . , n) be a sequence of n samples of the process taken at the time instants

Find the mean and the variance of the sample mean

Since X(t) is zero-mean and stationary, we have

and Rx(ti, tk) = E[X(ti)X(tk)] = RX(tk - ti) = Rx

Thus