Example 12.1 (AR(1))
The AR(1) process from Example
10.1 with

has the
characteristic equation

. The explicit solution

and

occurs exactly when

. The
inverse filter of

is thus

and the
MA(

) representation of the AR(1) process is
The ACF of the AR(1) process is

. For

all autocorrelations are positive, for

they
alternate between positive and negative, see Figure
11.3.
Example 12.2 (AR(2))
The AR(2) process with

,
is stationary when given the roots

and

of the
quadratic equation
it holds that

and

. We obtain solutions as
and

. Due to

and

it
holds that

and
 |
(12.11) |
From the Yule-Walker equations in the case of an AR(2) process
it follows that

. The case

is excluded because a root would lie on the unit circle
(at 1 or -1). Thus for a stationary AR(2) process it must hold
that
from which, together with (
11.11), we obtain the
`stationarity triangle'
i.e., the region in which the AR(2) process is stationary.
The ACF of the AR(2) process is recursively given with
(11.12), (11.13) and
for
. Figure
(11.4) displays typical patterns.