I am looking at a theorem about transformations of stationary processes that reads:
If $(e_t)_t$ is a strictly stationary process, and $x_t=f(e_t,e_{t-1},e_{t-2},...) \in \mathbb{R}^k$ is a random vector, then $(x_t)_t$ is a strictly stationary process.
It is emphasized in the main text that the transformation can include the full history of $e_t$
Counterexample: Suppose $(e_t)_t$ is some i.i.d. process (strictly stationary by design) and the proposed transformation is $x_t=\sum_{j=0}^t e_j$
$x_t$ is absolutely not a strictly stationary process?
How can both be true at the same time?
Thanks