Suppose we want to get a lower bound for the probability $$\mathbb{P}(X>a, Y<b),$$ and let us assume $\mathbb{P}(X>a) + \mathbb{P}(Y<b) <1$, so a direct union bound would not work.
If $X$ and $Y$ were independent, then we get a lower bound which is $\mathbb{P}(X>a)\cdot \mathbb{P}(Y<b)$. Now if we only know that $|\text{Corr}(X, Y)| < \epsilon_0$, can we show that for $\epsilon_0$ is sufficiently smaller, we can obtain some form of a lower bound for $\mathbb{P}(X>a, Y<b)$?
In my actual setup, we have a random walk $\{Z_i\}$ with $\rho$-mixing condition, and for two disjoint collection of indices $\mathcal{A}$ and $\mathcal{B}$, I would like to lower bound $$\mathbb{P}\Big(\sum_{i \in \mathcal{A}} Z_i > a, \sum_{j \in \mathcal{B}} Z_j <b\Big)$$ when the distance between $\mathcal{A}$ and $\mathcal{B}$ is large.