I am looking for a bound on the entropy $H(X+Y)$ of the sum of two independent discrete random variables $X$ and $Y$. Naturally, $$H(X+Y) \leq H(X) + H(Y) ~~~~~~(*)$$ However, applied to the sum of $n$ independent Bernoulli random variables $Z_1, \ldots, Z_n$, this gives $$ H(Z_1 + Z_2 + \cdots + Z_n) \leq n H(Z_1)  $$ In other words, the bound grows linearly with $n$ when applied repeatedly. However, $Z_1 + \cdots Z_n$ is supported on a set of size $n$, so its entropy is at most $\log n$. In fact, by the central limit theorem, I'm guessing that $H(Z_1 + \cdots + Z_n) \approx (1/2) \log n$ since it is essentially supported on a set of size $\sqrt{n}$. In short, the bound $(*)$ overshoots by quite a bit in this situation. From perusing this blog post, I gather all sorts of bounds on $H(X+Y)$ are possible; is there a bound that gives the right asymptotics (or, at least, more reasonable asymptotics) when applied repeatedly to the sum of Bernoulli random variables?
On the entropy of a sum
it.information theory;shannon entropy