Kyung Hee University
Random Processing
Department of Electronics and Radio Engineering
Prof. Hyundong Shin
Communications and Coding Theory Laboratory (CCTLAB)
XII-1
C1002900 RP Lecture Handout XII:
Stationary Processes
Reading:
Chapter 9.1
Any pair of samples of the process
Xt
with the same temporal separation, i.e.,
and
Xs
for any
, might have the same correlation.
This kind of “statistical
time-invariance
” which is termed
stationary
is in fact a ra-
ther natural property of many physical stochastic processes.
12.1. Strict-Sense Stationarity
The strongest notion of stationarity is referred to as
strict-sense
stationarity and is de-
fined in terms of the complete statistical characterization of the process. Specifically, a
stochastic process
is said to be
strict-sense stationary (SSS)
if all its finite-
dimensional distributions are time-invariant, i.e.,
,,
,
,
,
,
,,,
N
N
NN
Xt Xt
f
x
xxf
x
xx
12
1
2
(XII.1)
for all choices of
N
, time instants
N
tt
1
, and for any
.
The process
does not have absolute time reference, i.e., its statistical prop-
erties are invariant to a shift of the origin.
From the definition, it follows that
X
fx
f
x
0
.
For the second order,
,
,
x
f
x
x
f
x
x
1
2
where
. Hence, the joint density of
and
is independent
of
t
.
There are many examples of SSS random processes, e.g.,
diiscrete-time white Gaussian
noise process and random telegraph process.