Page 214 - Probability, Random Variables and Random Processes

RANDOM PROCESSES [CHAP 5

Since X(t) has stationary increments, we have
Var[X(t + h) - X(t)] = Var[X(h)] = a2h
in view of Eq. (5.63). Hence,
a2h
lim P{ I X(t + h) - X(t) I > E) = lim - = 0
h-0 h-rO gZ
Thus the Wiener process X(t) is continuous in probability.

Supplementary Problems

5.63. Consider a random process X(n) = {X,, n 2 l), where
x,=z, +z2 + -+z,
and Z, are iid r.v.'s with zero mean and variance a2. Is X(n) stationary?
Ans. No.

5.64. Consider a random process X(t) defined by
X(t) = Y cos(ot + 0)
where Y and 0 are independent r.v.3 and are uniformly distributed over (-A, A) and (- K, K), respectively.
(a) Find the mean of X(t).
(b) Find the autocorrelation function Rx(t, s) of X(t).
Ans. (a) E[X(t)] = 0; (b) Rx(t, s) = i~~ cos O(t - S)
5.65. Suppose that a random process X(t) is wide-sense stationary with autocorrelation
R,(t, t + z) = e-1'112

(a) Find the second moment of the r.v. X(5).
(b) Find the second moment of the r.v. X(5) - X(3).
Ans. (a) E[X~(~)] (b) E{[X(5) - x(3)I2) = 2(1 - e- ')
= 1 ;
5.66. Consider a random process X(t) defined by
X(t) = U cos t + (V + 1) sin t - co < t < cx,
where U and V are independent r.v.'s for which
E(U) = E(V) = 0 E(UZ) = E(V2) = 1
(a) Find the autocovariance function Kx(t, s) of X(t).
(b) Is X(t) WSS?
Ans. (a) Kx(t, s) = cos(s - t); (b) No.
5.67. Consider the random processes

where A,, A,, a,, and w, are constants, and r.v.3 0 and 0 are independent and uniformly distributed over
( - w, 4.