Suppose $X_1$ and $X_2$ are i.i.d. Binomial random variable where $X_1, X_2 \sim \mathrm{Bin}(n, 1/2)$. What can we say about the random variables $X_1 + X_2$ and $X_1 - X_2$?
I know they are not independent because only independent Gaussian random variables have independent sums and differences, but could they be "close to" independent? In particular, does there exist independent Binomial random variables $Y_1$ and $Y_2$ such that $Y_1 = X_1 + X_2 + O(1)$ and $Y_2 = X_1 - X_2 + O(1)$ (if we approximate Binomials with Gaussians then the error would be $O\left(\sqrt{n}\right)$ but maybe a better bound is possible)?