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Renyi entropy of a random pair $(X,Y)$ with probability distribution $p_{X,Y}$ is defined by \begin{equation} H_\alpha(X,Y) = \frac{1}{1-\alpha}\log\sum_{x,y} p_{X,Y}(x,y)^\alpha. \end{equation} Assume that $X$ and $Y$ have countably infinite alphabets, and observe the case $0\leq\alpha\leq 1$.

It is known that this functional is subadditive, i.e., that $H_\alpha(X,Y)\leq H_\alpha(X) + H_\alpha(Y)$, only for $\alpha=0$ (Hartley entropy) and $\alpha=1$ (Shannon entropy).

My question is the following: Does there exist some upper bound on the joint entropy for $0<\alpha<1$, in terms of the marginal entropies? In fact, I don't even need an explicit upper bound, answering the following simpler question would suffice: If $H_\alpha(X)<\infty$ and $H_\alpha(Y)<\infty$, does there exist a constant $h$ such that $H_\alpha(X,Y)\leq h$?

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Related:… – Piotr Migdal May 14 '13 at 15:33

The answer is no. Renyi entropy can even be unbounded and discontinuous over the set of probability distributions with given marginals (but not always, this depends on the marginals). You can find detailed proofs in

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