Here is a problem that just occurred to me and that may be novel or interesting, at least I could find no trace on here. I have a short proof by contradiction in mind but it suffers from a limiting issue.
Assume $\{X_t\}$ is a discrete time stochastic process. Define the stochastic process $\{Y_t\}$ as $Y_t:=X_tX_{t-1}$.
1. Question. If $\{Y_t\}$ is strictly stationary, can we conclude that $\{X_t\}$ must be strictly stationary as well?
2. Follow-on question. How would the answer change if instead $Y_t:=X_tX_{t-1}X_{t-2}\cdots$?