ON THE MAXIMUM-ENTROPY OF THE SUM OF 2 DEPENDENT RANDOM-VARIABLES

We investigate the maximization of the differential entropy h(X + Y) o f arbitrary dependent random variables X and Y under the constraints o f fixed equal marginal densities for X and Y. We show that max h(X + Y ) = h(2X), under the constraints that X and Y have the same fixed marg inal density f, if and only if f is log-concave. The maximum is achiev ed when X = Y. If f is not log-concave, the maximum is strictly greate r than h(2X). As an example, identically distributed Gaussian random v ariables have log-concave densities and satisfy max h(X + Y) = h(2X) w ith X = Y. More general inequalities in this direction should lead to capacity bounds for additive noise channels with feedback.