Page 170 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 51 RANDOM PROCESSES

A. Stationary Processes:
A random process {X(t), t E T) is said to be stationary or strict-sense stationary if, for all n and
for every set of time instants (t, E T, i = 1,2, . . . , n),

for any 2. Hence, the distribution of a stationary process will be unaffected by a shift in the time
origin, and X(t) and X(t + 2) will have the same distributions for any z. Thus, for the first-order
distribution,
FX(x; t) = FX(x; t + 2) = FAX)
and fAx; t) = fx(x)
Then Px(t) = ECX(t)l = P
Var[X(t)J = a2

where p and a2 are contants. Similarly, for the second-order distribution,
Fxh x2; t1, t2) = Fx(x1, x2; t2 - t1) (5.1 9)
and fx(% X2; tl, t2) =fx(~l, ~2; - tl) (5.20)
t2
Nonstationary processes are characterized by distributions depending on the points t,, t, , . . . , tn .

B. Wide-Sense Stationary Processes :
If stationary condition (5.14) of a random process X(t) does not hold for all n but holds for n 5 k,
then we say that the process X(t) is stationary to order k. If X(t) is stationary to order 2, then X(t) is
said to be wide-sense stationary (WSS) or weak stationary. If X(t) is a WSS random process, then we
have
1. E[X(t)] = p (constant)
2. Rx(t, S) = E[X(t)X(s)] = Rx( ( s - t 1 )
Note that a strict-sense stationary process is also a WSS process, but, in general, the converse is not
true.

C. Independent Processes:
In a random process X(t), if X(ti) for i = 1,2, . . . , n are independent r.v.'s, so that for n = 2,3, . . . ,
n
FJX~, . . . , xn; t,, . . . , t,) = n F~(X,; ti)
i= 1
then we call X(t) an independent random process. Thus, a first-order distribution is sufficient to charac-
terize an independent random process X(t).

D. Processes with Stationary Independent Increments:
A random process {X(t), t 2 0) is said to have independent increments if whenever 0 < t, < t, <
... < t,,

X(O), X(t1) - X(O), X(t2) - X(tl), . . X(tn) - X(tn- 1)