Stat 4870/7870: Introduction to Time Series
Chapter 2: Correlation and Stationary Time Series
Section 2: Stationarity
Definition 0.1
(Strictly Stationary)
.
A strictly stationary time series is one
for which the probabilistic behavior of every collection of values and shifted
values
{
x
i
1
, x
i
2
, . . . , x
i
k
}
and
{
x
i
1
+
h
, x
i
2
+
h
, . . . , x
i
k
+
h
}
,
are identical, for all
k
= 1
,
2
, . . .
, all time points
t
1
, t
2
, . . . , t
k
and all time
shifts
h
= 0
,
±
1
,
±
2
, . . .
.
Stationarity
•
It is difficult to assess strict stationarity from data, however, station
ary time series data should exhibit similar behaviors over different time
intervals.
•
Instead of imposing conditions on all possible distributions of a time
series, we will use a milder version that imposes conditions only on the
first two moments.
Definition 0.2
(Weakly Stationary)
.
A weakly stationary time series is a
finite variance process where
(
i
) the mean value function,
μ
t
is constant and does not depend on time
t
,
and
(
ii
) the autocovariance function
γ
(
s, t
) depends on
s
and
t
only through their
distance

s

t

.
Remark.
On stationary:
1