Suppose we have a continuous-time stochastic process $s(t)$ that's mixing. What properties would a function $f$ have to have so that $f(s(t))$ would also be mixing?


I'm sure this is a well-known thing, I'm just having trouble locating a term for such functions or any relevant literature.



Functions that preserve the mixing of a stochastic process
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Let $X:=(X\_t)\_{t\in\R}$ be a stochastic process in a state space $S$, endowed by a sigma-algebra $\S$, so that, for each $t\in\R$, $X\_t$ is a random variable (r.v.) on a probability space $(\Omega,\F,\P)$ with values in $S$.
The [strong mixing coefficient](https://en.wikipedia.org/wiki/Mixing_(mathematics)#Mixing_in_stochastic_processes) for $X$ is 
$$a\_X(s):=\sup\{|\P(A\cap B)-\P(A)\P(B)|\colon t\in\R,A\in X\_{-\infty}^t,B\in X\_{t+s}^\infty\},
$$
where $s>0$ and $X\_u^v$ is the sigma-algebra generated by the set $\{X\_t\colon t\in\R, u\le t\le v\}$ of r.v.'s. 
Assume that the process $X$ is strongly mixing, i.e., $a\_X(s)\to0$ as $s\to\infty$. 


Let $(T,\T)$ be any measurable space. For each $t\in\R$, let $Y\_t:=f(X\_t):=f\circ X\_t$, where $f\colon S\to T$ is a measurable function. 


Then $Y:=(Y\_t)\_{t\in\R}$ is a strongly mixing stochastic process in the state space $T$. This follows immediately, because $Y\_u^v\subseteq X\_u^v$ for all $u$ and $v$ in $[-\infty,\infty]$ and hence $0\le a\_Y(s)\le a\_X(s)$ for all real $s>0$. 


Thus, the strong mixing is preserved under any measurable transformation $f$. 


