# Upper bound on joint Renyi entropy

Renyi entropy of a random pair $(X,Y)$ with probability distribution $p_{X,Y}$ is defined by $$H_\alpha(X,Y) = \frac{1}{1-\alpha}\log\sum_{x,y} p_{X,Y}(x,y)^\alpha.$$ 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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–  Piotr Migdal May 14 '13 at 15:33