Page 172 - Probability, Random Variables and Random Processes

RANDOM PROCESSES [CHAP 5

are independent. If {X(t), t 2 0) has independent increments and X(t) - X(s) has the same distribu-
tion as X(t + h) - X(s + h) for all s, t, h 2 0, s < t, then the process X(t) is said to have stationary
independent increments.
Let {X(t), t 2 0) be a random process with stationary independent increments and assume that
X(0) = 0. Then (Probs. 5.21 and 5.22)

where p, = E[X(l)] and

where a12 = Var[X(l)].
From Eq. (5.24), we see that processes with stationary independent increments are nonstationary.
Examples of processes with stationary independent increments are Poisson processes and Wiener
processes, which are discussed in later sections.

E. Markov Processes:

A random process (X(t), t E 7') said to be a Markov process if
is

whenever t1 < t2 < < t, < t,,,.
A discrete-state Markov process is called a Markov chain. For a discrete-parameter Markov
chain {X,, n 2 0) (see Sec. 5.5), we have for every n
P(X,+, = j(X, = i,, Xi = i,, ..., Xn = i) = P(Xn+, = jlXn = i) (5.27)
Equation (5.26) or Eq. (5.27) is referred to as the Markov property (which is also known as the
memoryless property). This property of a Markov process states that the future state of the process
depends only on the present state and not on the past history. Clearly, any process with independent
increments is a Markov process.
Using the Markov property, the nth-order distribution of a Markov process X(t) can be
expressed as (Prob. 5.25)

p{X(tk) 2 ~k) I X(tk - 1) = xk - I) (5.28)

Thus, all finite-order distributions of a Markov process can be expressed in terms of the second-order
distributions.

F. Normal Processes :
A random process {X(t), t E T) is said to be a normal (or gaussian) process if for any integer n
and any subset (t,, . . ., t,) of T, the n r.v.'s X(tl), ..., X(t,) are jointly normally distributed in the
sense that their joint characteristic function is given by

where w,, . .., on are any real numbers (see Probs. 5.59 and 5.60). Equation (5.29) shows that a
normal process is completely characterized by the second-order distributions. Thus, if a normal
process is wide-sense stationary, then it is also strictly stationary.