In this paper we investigate the capacity regions of fading broadcast channels with memory, and only statistical channel state information at the transmitter. Under this setting, the ergodic capacity region is open in general. Instead of match- ing achievable schemes and upper bounds, in this work we resort to classifying the random channels through distributions for which we can achieve the capacity results. In particular, we derive sufficient conditions to attain the degradedness by stochastic ordering with the same marginal property. We also illustrate examples to show the scenarios of the derived conditions.