I have the lemma that for any measurable function and $\alpha$-mixing process $\{x_t\}$, where $x_t$ can be a sequence of random vectors or just univariate random variables. Then $\{f(x_t)\}$, is also an $\alpha$-mixing process.
Now if I have a stochastic process of p-dimensional random vectors $\{\mathbf{x}_t\}$, where $\mathbf{x}_t = (x_{1t}, \dots , x_{pt})^T$, that is an $\alpha$-mixing process. Can I make the following statements using the previous lemma:
- A single r.v $\{x_{it}\}$ is alpha mixing, using the lemma with $f(x_1, \dots x_p) = x_i$
- The product $\{x_{it} x_{jt}\}$ is alpha mixing, using the lemma with $f(x_1, \dots x_p) = x_i x_j$
This is indeed true. However, I would rather say that the process $\{x_{i,t},t\geqslant 1\}$ is $\alpha$-mixing instead of a single random variable.