# Is the output of function of two ergodic processes ergodic?

Let $\{\xi_k\}_{k\in \mathbf{Z}}$ and $\{\epsilon_k\}_{k\in \mathbf{Z}}$ be two independent zero-mean Gaussian processes (i.i.d.). Is the output of the function $f$ such that $y = f(\dots,\xi_{k-1},\xi_{k},\xi_{k+1},\dots,\epsilon_{k-1},\epsilon_k,\epsilon_{k+1},\dots)$ is an ergodic process? I found similar claims in the book ("Stochastic Processes and Long Range Dependence"). Can anybody show me some references or guide me through mathematical language that is easy to understand? Thank you!

• You may need some conditions on f. E.g. if f accumulates one of the input signals, then the output may not even be stationary. – Juancho Jan 15 '18 at 16:02
• Thank you for your comment @Juancho. What did you mean by "accumulate"? Does it mean accumulate the moments? I also have some other questions. Is the linear combination of two independent ergodic stochastic processes an ergodic process? This seems promising but I can not prove it yet. – ZHUANG Jan 16 '18 at 1:56