Given a sequence of dependent random variables $(X_i)_i$, what are some ways to measure the amount of dependence in this process?
The only clear example of this that I can refer to would be to couple the $X_i$'s with a sequence of independent variables $Y_i$ such that the coupling satisfies some constraints (as in the following example). For example, let $N=\prod_{p\le n}p^{C_p}$ be a uniform integer from $1$ to $n$; this necessarily implies that the prime power process $(C_p)_{p\le n}$ consists of dependent variables. Define $M=\prod_{p\le n}p^{Z_p}$, where the variables $Z_p \ge 0$ are independent and $\text{Geometric}\left(1/p\right)$. Arratia's conjecture (page 17 of https://arxiv.org/pdf/1305.0941.pdf) is an attempt to measure the amount of dependence in the dependent process $(C_p)_p$ by coupling $N, M$ such that $$N \text{ divides } MP$$ for some random prime $P\le n$. [To see that the divisibility constraints concerns the $C_p$'s and $Z_p$'s, rewrite the divisibility relation as $\sum_{p\le n}(C_p - Z_p)^+ \le 1$), where $(\cdot)^+$ denotes the positive part.]
A good starting point, with references, is
[1] https://www.stat.umn.edu/geyer/8112/notes/stationary.pdf
In particular on page 6 you will find comparison of some (terribly named) standard notions:
phi-mixing implies beta-mixing implies alpha-mixing implies mixing.
See Theorem 4 there for connections to the CLT.
For a detailed study see
[2] Introduction to Strong Mixing Conditions Volumes 1,2 and 3 by Richard C. Bradley. Published by Kendrick Press (January 1, 2007)