Assume that the sequence $\{X_i\}_{i\in\mathbb N^*}$ of random variables on $(\Omega,\mathcal A,P)$ is strongly mixing. Denote by $\mathcal F_i^k=\sigma(X_t, i\leq t\leq k)$ a sigma-algebra generated. As commented on this post, for the sets $A_{n,i}=\{\omega:0\leq X_i(\omega)\leq b_n\}$ and $A_{n,j}=\{\omega:0\leq X_j(\omega)\leq b_n\}$ where $b_n$ is a postive real sequence converging to zero, we have that $$P(A_{n,i})P(A_{n,j})-\alpha(\lvert i-j \rvert)\leq P(A_{n,i}\cap A_{n,j})\leq P(A_{n,i})P(A_{n,j})+\alpha(\lvert i-j \rvert).$$
I realized that, given a mixing rate (for example, the arithmetic rate $\alpha (k)\leq C k^{-d}$), if $\lvert i-j\rvert>a_n^{-1}$ where $a_n$ is another positive sequence converging to zero, then $\alpha(\lvert i-j\rvert)\leq Ca_n^d$ since $\alpha(\cdot)$ is nonincreasing. In my case, $a_n^d=o(P(A_{n,i})P(A_{n,j})),n\to\infty.$ Hence we conclude that there are constants $c_1,c_2>0$ such that for all $n$ sufficiently large $$c_1P(A_{n,i})P(A_{n,j})\leq P(A_{n,i}\cap A_{n,j})\leq c_1P(A_{n,i})P(A_{n,j}),$$ which is totally consistent with the intuition that "mixing" is a notion of asymptotic independence. That is, when $\lvert i-j\rvert$ is "large", $P(A_{n,i}\cap A_{n,j})\approx P(A_{n,i})P(A_{n,j})$.
However, little is known about the relation between $P(A_{n,i})P(A_{n,j})$ and $P(A_{n,i}\cap A_{n,j})$ when the lag is "short" $\lvert i-j\rvert\leq a_n^{-1}$.
Problem
I'm providing lower bounds for integrals of the form $$\iint_{[0,1]^2} P(tb_n\leq X_i\leq b_n,zb_n\leq X_j\leq b_n)f(t,z)dtdz$$ which is hard to deal with, where $f$ is some function. However, $$\iint_{[0,1]^2} [P(tb_n\leq X_i\leq b_n)P(zb_n\leq X_j\leq b_n)]^\theta f(t,z)dtdz$$ is much easier, where $\theta>0$. I need to bound the integral for all $i,j\in\{1,\dotsc,n\}$, and then I thought of assume that
\begin{equation} P(tb_n\leq X_i\leq b_n,zb_n\leq X_j\leq b_n)= \begin{cases} c_{i,j}[P(tb_n\leq X_i\leq b_n)P(zb_n\leq X_j\leq b_n)]^\theta &,\lvert i-j\rvert\leq a_n^{-1}\\ c'_{i,j} P(tb_n\leq X_i\leq b_n)P(zb_n\leq X_j\leq b_n)&,\lvert i-j\rvert> a_n^{-1} \end{cases} \end{equation} for some constants $c_{i,j},c'_{i,j}>0$, some $\theta\in(0.5,1.5)$ and all $n$ large enough (I'm working with asymptotic theory). The number $\theta\in(1/4,3/4)$ tells that when the $\lvert i -j\rvert$ is short, $P(tb_n\leq X_i\leq b_n,zb_n\leq X_j\leq b_n)$ is slightly smaller or larger than $P(tb_n\leq X_i\leq b_n)P(zb_n\leq X_j\leq b_n)$. If it is too strong, we can change the exponent by $\theta_{i,j}$ such that $\theta_{i,j}\in(1/4,3/4)$.
Do you think this hypothesis is reasonable? Is there any contradiction in assuming it in conjunction with strong mixing dependence?
I appreciate any feedback or advice. Thanks.