The capacity region of a class of two-user degraded compound broadcast channels

Weingarten et al. established the capacity region for the two-user degraded compound broadcast channel where the realizations of each user exhibit a certain degradedness order. The degradedness order is defined through an additional fictitious user whose channel is stochastically degraded with respect to each realization from one set (the stronger receiver) while each realization from the other set (the weaker receiver) is stochastically degraded with respect to him. Rather than specifying a fictitious user, we consider a two-user degraded compound broadcast channel which is pair-wise degraded, i.e., each realization from the weaker receiver is stochastically degraded with respect to each realization from the stronger receiver. In this paper, we first prove the capacity region for a discrete memoryless class of this broadcast channels where the weaker receiver has only two possible realizations while there is an arbitrary number of possible realizations for the stronger receiver. Next, we consider the equivalent class of degraded compound MIMO Gaussian broadcast channels. To show that Gaussian inputs attain the capacity region, we prove a new extremal entropy inequality using a technique recently introduced by Geng and Nair that was used to resolve the capacity region of the two-user MIMO Gaussian broadcast channel with common and private information.