If $X_t$ and $Y_t$ are independent random processes that are $\alpha$-mixing, is a linear combination, $aX_t + bY_t$ also $\alpha$-mixing? What about other functions $f(X_t,Y_t)$? How does one approach a problem like this?
In particular I am interested in if $\mathbf{Z}_t$ is $\alpha$-mixing where $Z_{t,i} = \delta_{i,X_t} - \delta_{i,Y_t}$ where $\delta_{ij}$ is Kronecker's Delta $(i=1,\ldots,k)$
Notice that if $f\colon\mathbf R\to\mathbf R$ is Borel-measurable, then for each $I\subset\mathbb Z$, $\sigma\left(f(X_t,Y_t),t\in I\right)\subset \sigma\left((X_t,Y_t),t\in I\right)$. Therefore, it suffices to estimate the mixing coefficients of the sequence $\left((X_t,Y_t)\right)_{t\in\mathbb Z}$. By approximation, it suffices to estimate for a fixed $n$ the quantity $$\left|\mathbb P\left(\bigcap_{j=-m}^0\{(X_j,Y_j)\in B_j\}\cap \bigcap_{j=n}^{n+m}\{(X_j,Y_j)\in B_j\}\right)-\mathbb P\left(\bigcap_{j=-m}^0\{(X_j,Y_j)\in B_j\}\right)\mathbb P\left(\bigcap_{j=n}^{n+m}\{(X_j,Y_j)\in B_j\}\right)\right|,$$ where $B_j$ is a Borel subset of $\mathbb R^2$ which can be written as a finite disjoint union of products of Borel subsets of the real line. Using independence, we can show that $$\alpha\left(\left(\left(X_t,Y_t\right)\right)_{t\in\mathbb Z},n\right) \leqslant \alpha\left(\left(X_t\right)_{t\in\mathbb Z},n\right)+ \alpha\left(\left(Y_t\right)_{t\in\mathbb Z},n\right).$$