I am having problems with computing alpha-mixing rate in a sequence of random variables as the notation and definition seems a bit abstract to me.
For example, suppose that $Y_i = \sum_{k=1}^i V_i$ where $V_i$ are i.i.d. standard normal random variables. How can I determine if this sequence is strongly mixing? What is the mixing rate $\alpha_i$ of the sequence $\{Y_1, Y_2, \dots \}$?
Denote by $\sigma$ the $\sigma$-algebra generated by a random variable or a collection of random variables. Since (by definition of the mixing coefficients) $$ \alpha_i=\sup_{j}\alpha\left(\sigma\left(Y_{k},1\leqslant k\leqslant j\right),\sigma\left(Y_{\ell},j+i\leqslant \ell\right)\right), $$ the following inequality holds: $$ \alpha_i\geqslant \alpha\left(\sigma\left(Y_{i}\right),\sigma\left(Y_{2i}\right)\right)\geqslant \alpha\left(\sigma\left(i(U+V)\right),\sigma\left(iU\right)\right) ,$$ where $U$ and $V$ are independent and have standard normal distribution. Since $\sigma\left(i(U+V)\right)=\sigma\left(U+V\right)$ and $\sigma\left( iU\right)=\sigma\left( U\right) $, it follows that $$ \alpha_i\geqslant \alpha\left(\sigma\left( U+V \right),\sigma\left( U\right)\right)>0 $$ due to the fact that $U+V$ and $U$ are not independent, hence the sequence $(Y_k)_{k\geqslant 1}$ is not $\alpha$-mixing.