Criterions for strong mixing

Question

I just read about $\alpha$ (or strong) mixing (as defined here http://arxiv.org/pdf/math/0511078.pdf on pages 2 and 5). Assume, I've some random variables $(X_i)_{i \geq 1}$ which are not independent (for example $X_i:=Y_i \times \frac{\sum\limits_{i=1}^{N}Y_i}{N}$ for $(Y_i)_{i \geq 1}$ i.i.d and some $N>0$). How can I prove that the $(X_i)_{i \geq 1}$ are strong-mixing and compute the $\alpha$-coefficients (or show that they are not strong-mixing)? Are there any criterions besides the formal definition to show strong mixing?

Answer 1

Usually, checking that a process is $\alpha$-mixing or not is a hard task. Even linear processes with Bernoulli innovations $(e_i)$, namely, $X_i=\sum_{l\geqslant 0}2^{-l}e_{i-l}$ are not $\alpha$-mixing.

However, sometimes, we can get a bound on mixing coefficients if the studied process is built from countable many independent process (see Theorem 5.2. in the paper by Richard C. Bradley linked in the question).

Moreover, using Theorem 3.1. of the same paper, we have criteria for determining whether a functional of a Markov chain is mixing or not.