I would like to apply a CLT theorem to a sequence of random variables $X_1,X_2,X_3, \ldots $ which are "almost independent". Specifically, I want to apply a CLT theorem that replaces the usual stochastic independence requirement with mixing as discussed in [1,2]. A permissive requirement on the mixing coefficients $\alpha_n$ seems to be the existence of some $\delta > 0$ such that
$\sum_{n = 1}^\infty \alpha_n^{\frac{\delta}{2(2+\delta)}} < \infty$
Since one has $\alpha_n \to 0$ as $n \to \infty$, the above condition requires essentially that $\alpha^n \leq C n^{-\eta}$ for some $\eta > 2$ and $C \geq 0$.
Question: Are there any known versions of the CLT theorem which would get away with some $1 < \eta \leq 2$? Any help or reference would be appreciated.
[1] https://en.wikipedia.org/wiki/Central_limit_theorem#CLT_under_weak_dependence
[2] https://en.wikipedia.org/wiki/Mixing_(mathematics)#Mixing_in_stochastic_processes
A sufficient condition for the central limit theorem for $\alpha$-mixing sequences is that $(X_i)$ is stationary and
$$ \tag{DMR} \int_0^1\alpha^{-1}(u)Q(u)^2du<\infty, $$ where $\alpha^{-1}(u)=\operatorname{Card}\{k,\alpha(k)\geqslant u\}$ and $Q(u)=\inf\{t>0,\mathbb P(\lvert X_1\rvert >t)\leqslant u\}$. This is done in Theorem 1 of
Doukhan, Paul, Massart, Pascal, and Rio, Emmanuel. "The functional central limit theorem for strongly mixing processes." Annales de l'I.H.P. Probabilités et statistiques 30.1 (1994): 63-82. http://eudml.org/doc/77475.
In particular, if $\mathbb E\left[\lvert X_0\rvert^{2+\delta}\right]$ is finite for some $\delta$, then (DMR) is satisfied as long as $\sum_{n\geqslant 1}n^{1/(1+\delta)}\alpha_n$.
Moreover, condition (DMR) has been shown in the aforementioned paper to be optimal.