Given a random variable $X$ distributed under some computable distribution $P$ we have,
$$0 \le E[K(X)] - H(P) \le K(P)$$
Where $H(P)$ is the entropy of $P$. I tried using Hoeffding concentration bounds, but they aren't very helpful here. Are there similar bounds for $E[(K(X)-H(P))^2]$? In general, is there any way to get information about the distribution shape of $K(X)$ or is that uncomputable?