Finite-size correction and variance of the mutual information of random linear estimation with non-Gaussian priors: A replica calculation

Random linear vector channels have been known to increase the transmission of information in several communications systems. For Gaussian priors, the statistics of a key metric, namely, the mutual information, which is related to the free energy of the system, have been analyzed in great detail for various types of channel randomness. However, for the realistic case of non-Gaussian priors, only the average mutual information has been obtained in the asymptotic limit of large channel matrices. In this paper, we employ methods from statistical physics, namely, the replica approach, to calculate the finite-size correction and the variance of the mutual information with non-Gaussian priors, both for the case of correlated Gaussian and uncorrelated non-Gaussian channel matrices in the same asymptotic limit. Furthermore, using the same methodology, we show that higher order cumulants of the mutual information should vanish in the large-system-size limit. In addition, we obtain closed-form expressions for the minimum mean-square error finite-size corrections and variance for both Gaussian and non-Gaussian channels. Finally, we provide numerical verification of the results using numerical methods on finite-sized systems.