Upper bound on joint Renyi entropy - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T18:32:00Z http://mathoverflow.net/feeds/question/111466 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111466/upper-bound-on-joint-renyi-entropy Upper bound on joint Renyi entropy newuser 2012-11-04T15:35:02Z 2013-05-14T15:25:53Z <p>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 $0\leq\alpha\leq 1$. </p> <p>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). </p> <p>My question is the following: Does there exist some upper bound on the joint entropy for $0&lt;\alpha&lt;1$, 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 $H_\alpha(X)&lt;\infty$ and $H_\alpha(Y)&lt;\infty$, does there exist a constant $h$ such that $H_\alpha(X,Y)\leq h$?</p>