Nonstationary processes 2: here is a second example that provides some ground for intuition.
Example 2. Random walk. Consider
where is white noise:
(2) , for all and for all
(1) is an example of a dynamic model. It does not allow us to investigate properties of directly because there is a reference to the process at moment , which is an unknown itself. To get rid of , we use the procedure called recurrent substitution. (1) is assumed to hold for all , so for the previous period it looks like this:
Plugging (3) in (1) we get After doing this times we obtain
In Example 1 the range of time moments didn't matter because the model wasn't dynamic. Here we have to assume that in (1) takes all positive integer values and in (4) is some positive integer. Then can be taken equal to , so that
As , the initial value doesn't matter much. For simplicity, we assume that it is zero. Then we see that (1) implies
This representation is free from references to unknown variables, and can be easily used to study the properties of the process under consideration. For example, the first condition of a stationary process is satisfied: . However, the second is violated (use (2)):
which changes with . Thus, under our assumptions (1) is an example of a nonstationary process.