If $X_i$ be a $\alpha$-mixing séquence, what about $X_i^2$?

Question

Let $(X_i)_{i\in \mathbb{Z}}$ be an $\alpha$-mixing sequence of random variable.

Is the sequence $(X_i^2)_{i\in\mathbb Z}$ also an $\alpha$-mixing sequence?

Answer 1

More generally, if $h\colon\mathbb R\to \mathbb R$ is a Borel-measurable function and the sequence $X:=(X_i)_i$ is $\alpha$-mixing, so is $Y:=(h(X_i))_i$. Indeed, if $m$ and $n$ are fixed integers, then $$\sigma(h(X_j),j\leqslant m)\subset\sigma(X_j,j\leqslant m)\mbox{ and }\sigma(h(X_j),j\geqslant m+n)\subset\sigma(X_j,j\geqslant m+n),$$ which gives $\alpha_Y(n)\leqslant \alpha_X(n)$ for each integer $n$.