Let $\{X_t\}_{t=1}^\infty$ be a stationary $\alpha$-mixing process, where $X_t \in \mathbb{R}^m$ and let $f:\mathbb{R}^m \rightarrow \mathbb{R}^n $ be a Borel-measurable function. Is the transformed process $\{f(X_t)\}_{t=1}^\infty$ is $\alpha$-mixing as well?
If so what is the relation between the two mixing coefficients?
Is it also true for other notions of mixing? ($\beta, \phi$) ?
The $\sigma$-algebra generated by $f(X_t)$ is contained in the $\sigma$-algebra generated by $X_t$; consequently for all $I\subset \mathbb Z$ (possibly infinite), $$ \sigma\left( f(X_t),t\in I\right)\subset \sigma\left( X_t,t\in I\right) $$ For this reason, for all $n$, $$ \alpha\left( \left( f(X_t)\right)_{t\geqslant 1},n\right)\leqslant \alpha\left( \left( X_t\right)_{t\geqslant 1},n\right) $$ and a similar relation holds for $\beta$ and $\phi$-mixing coefficients.