Along with examples of nonstationary processes, it is necessary to know a couple of examples of stationary processes.
Example 1. In the model with a time trend, suppose that there is no time trend, that is, . The result is white noise shifted by a constant , and it is seen to be stationary.
Example 2. Let us change the random walk slightly, by introducing a coefficient for the first lag:
where is, as before, white noise:
(2) , for all and for all
This is an autoregressive process of order 1, denoted AR(1).
Stability condition. .
By now you should be familiar with recurrent substitution. (1) for the previous period looks like this:
Plugging (3) in (1) we get After doing this times we obtain
To avoid errors in calculations like this, note that in the product the sum of the power of and the subscript of is always .
Here the range of time moments didn't matter because the model wasn't dynamic. In the other example we had to assume that in (1) takes all positive integer values. In the current situation we have to assume that takes all integer values, or, put it differently, the process extends infinitely to plus and minus infinity. Then we can take advantage of the stability condition. Letting (and therefore ) we see that the first term on the right-hand side of (4) tends to zero and the sum becomes infinite:
We have shown that this representation follows from (1). Conversely, one can show that (5) implies (1). (5) is an infinite moving average, denoted MA().
It can be used to check that (1) is stationary. Obviously, the first condition of a stationary process is satisfied: . For the second one we have (use (2)):
which doesn't depend on .
Exercise. To make sure that you understand (6), similarly prove that
Without loss of generality, you can assume that . (7) is a function of the distance in time between , as required.