In a recent effort to understand stationary ergodic processes, I stumbled upon a paper that leaves me somewhat puzzled. I would be very grateful for any pointers.
The line of reasoning is as follows:
Given a set of assumptions, it is deduced that a random sequence $\{f_t:t\in\mathbb{Z}\}$ is stationary ergodic. For any measurable map $h$, stationary ergodicity of $\{f_t:t\in\mathbb{Z}\}$ implies that $\{h(f_t):t\in\mathbb{Z}\}$ is stationary ergodic.
Moreover, there is a sequence of innovations $\{u_t:t\in\mathbb{Z}\}$ - also presumed stationary ergodic.
Given these facts, the paper goes on to assert (verbatim)
Together with $\{u_t\}$ being SE (Assumption 3), it follows that $\{(u_t,h(f_t))\}$ is a stationary and ergodic vector sequence.
The proof concludes with the insight that: Continuity, and thus by extension, measurability of another function $g\,:\,\mathbb{R}^2\to\mathbb{R}$ in turn implies that $\{g(u_t,h(f_t))\}$ is stationary ergodic.
At this point my struggle is with the statement that “$\{(u_t,h(f_t))\}$ is a stationary and ergodic vector sequence”. While it was shown that $\{u_t\}$ and $\{f_t\}$ are stationary ergodic, I do not see how that necessarily implies that $\{(u_t,h(f_t))\}$ is jointly stationary ergodic. However, I am under the impression that this is a requirement for $\{g(u_t,h(f_t))\}$ to be stationary ergodic.
I would very much appreciate if anyone could tell me where I am going wrong, or wether I am missing any crucial piece of information - such as $\{u_t\}$ is assumed to be iid (?).
Thank you so very much.
Best,
Jon