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Using large-dimensional random matrix theory (RMT), we conduct mutual information analysis of a multiple-input multiple-output (MIMO) multiple access channel (MAC). Our channel model reflects the characteristics in small-cell networks where antenna correlations, line-of-sight components, and general type of fading distributions have to be included. The mutual information expression can be expressed as functionals of the Stieltjes transform through the so-called Shannon transform. Ideally, if the Stieltjes transform is known in the context of the large-dimensional RMT, then the problem is solved. However, it is difficult to derive the Stieltjes transform of the considered channel models directly, especially when the transmit correlation matrices are generally nonnegative definite and the channel entries are non-Gaussian. To overcome this, we use the generalized Lindeberg principle to show that the Stieltjes transforms of this class of random matrices with Gaussian or non-Gaussian independent entries coincide in the large-dimensional regime. This result permits to derive the deterministic equivalents (e.g., the Stieltjes transform and the ergodic mutual information) for non-Gaussian MIMO channels from the known results developed for Gaussian MIMO channels. As an application, we determine the capacity-achieving input covariance matrices for the MIMO-MACs and prove that the capacity-achieving input covariance matrices are asymptotically independent of the fading distribution.