Given a sequence $\{X_n\}_{n=1}^\infty$, suppose there exists a function $\alpha$ such that for each $n$ and $N$, \begin{equation} \sup \left\{ |P(A\cap B) - P(A)P(B)| : A \in \mathcal{F}_{X_1, \dots X_n}, B\in \mathcal{F}_{X_{n+k}, \dots, X_{N}}\right\} \leq \alpha(k). \end{equation} If $\alpha(k)\to 0$ as $k\to \infty$, then we say the sequence is strong mixing.
I am interested in the following statement which uses the covariance: there exists a function $\beta$ such that for each sequence of real numbers$\{a_i\}$, each integers $k\geq 1$, $n$, $N$ \begin{equation} \text{Cov}\Big(\log \Big(\sum_{i=1}^n a_i \prod_{k=1}^i X_k\Big), \log \Big(\sum_{j=n+k}^N a_j \prod_{l=n+k}^j X_l\Big)\Big) \leq \beta(k). \end{equation}
Is there any connection between the $\alpha$ and $\beta$ defined above? For example, does the $\alpha$ bound imply the $\beta$ bound? If not, what would be a stronger assumption for $\alpha$ so that $\beta$ holds?
If you can provide uniform moment bounds $$\mathbb{E}\left[\log\left(\sum_{i=1}^n a_i \prod_{k=1}^i X_k\right)^p\right]^{1/p} \le C_1; \qquad \mathbb{E}\left[\log\left(\sum_{j=n+k}^N a_j \prod_{l=n+k}^j X_l\right)^q\right]^{1/q} \le C_2$$ for some $p, q \ge 1$ such that $1 - \frac 1p - \frac 1q > 0$, then Davydov's inequality (see e.g. a previous question or thm. 1.2.3 in the book Mixing by Doukhan (1994) for a reference) implies you can take $$\beta(k) = 8 C_1 C_2 \alpha(k)^{1 - 1/p - 1/q}.$$