Consider two identically distributed independent random vectors $X$ and $Y$ of dimension $n$ (assume $n$ is large). Both $X$ and $Y$ have only positive integer entries in some finite range. I am trying to understand what conditions are sufficient for $H(X) + H(Y) \approx H(X+Y)$?
Is it possible to say what properties of the random distributions are sufficient for this to be true?
You can say that $$ H(X+Y)=H(X+Y | X) + H(X) - H(X | X+Y)=H(Y)+H(X)-H(X|X+Y), $$ so $H(X+Y) \approx H(X) + H(Y)$ iff $H(X|X+Y)\approx 0$, i.e., if you can essentially determine what $X$ must be when given $X+Y$. For instance, if the entries of $Y$ are always even, and the entries of $X$ can only be zero or one, then $H(X+Y)=H(X)+H(Y)$ exactly.