Theory of Statistics - George Mason University

124 1 Probability Theory
Definition 1.56 ((weakly) stationary process)
Suppose
{Xt : t = . . ., −2, −1, 0, 1, 2, .. .}
is such that E(Xt) = µ and V(Xt) < ∞ ∀ t and γ(s, t) is constant for any
fixed value of |s − t|. Then the process {Xt} is said to be weakly stationary.
A white noise is clearly stationary.
In the case of a stationary process, the autocovariance function can be
indexed by a single quantity, h = |s − t|, and we often write it as γh.
It is clear that in a stationary process, V(Xt) = V(Xs); that is, the variance
is also constant. The variance is γ0 in the notation above.
Just because the means, variances, and autocovariances are constant, the
distributions are not necessarily the same, so a stationary process is not necessarily
homogeneous. Likewise, marginal distributions being equal does not
insure that the autocovariances are constant, so a homogeneous process is not
necessarily stationary.
The concept of stationarity can be made stricter.
Definition 1.57 (strictly stationary process)
Suppose
{Xt : t = . . ., −2, −1, 0, 1, 2, .. .}
is such that for any k, any set t1, . . ., tk, and any h the joint distribution of
is identical to the joint distribution of
Xt1, . . ., Xtk
Xt1+h, . . ., Xtk+h.
Then the process {Xt} is said to be strictly stationary.
A strictly stationary process is stationary, but the converse statement does
not necessarily hold. If the distribution of each Xt is normal, however, and if
the process is stationary, then it is strictly stationary.
As noted above, a homogeneous process is not necessarily stationary. On
the other hand, a strictly stationary process is homogeneous, as we see by
choosing k = 1.
Example 1.31 a central limit theorem for a stationary process
Suppose X1, X2, . . . is a stationary process with E(Xt) = µ and V(Xt) = σ 2 .
We have
V( √ n(X − µ)) = σ 2 + 1
n
= σ 2 + 2
n
n
i=j=1
Cov(Xi, Xj)
n
(n − h)γh. (1.263)
h=1
Theory of Statistics c○2000–2013 James E. Gentle