In this paper, ergodic capacities of fading Gaussian multi-user channels with only statistical channel state information at the transmitter are considered. The main contributions are twofold: First, we introduce a framework to classify random fading channels solely based on their joint distributions, such that we can construct equivalent channels in order to directly make use of existing capacity results. To attain this goal, we leverage three schemes: maximal coupling, coupling, and copulas in addition to the usual stochastic order with the same marginal property. Second, we apply the developed framework to Gaussian broadcast channels, Gaussian interference channels, and Gaussian wiretap channels to obtain novel capacity (region) results.