Suppose we have stationary processes $X_1(t), X_2(t),..., X_n(t)$ and let $f_t(X_1(t), X_2(t),..., X_n(t))$ be a continuous function of these stationary processes. Will $f_t(\cdot)$ also be stationary in general?
PS. Not home work, for understanding only.
Thank you.
$X(n)$ be iid sequence. Let $Y(n)=X(1)$. Then both $X(\cdot)$ and $Y(\cdot)$ are stationary processes. Take $f(x,y)=xy$. This is a continuous function of $x$ and $y$. But $f(X(1),Y(1))=X(1)^2$ and $f(X(2),Y(2))=X(2)X(1)$ and these have different distributions.
(Example is on page 2 of http://econ.ucsb.edu/~doug/241b/Lectures/10%20Ergodic%20Stationarity.pdf)
