New Proofs of Gaussian Extremal Inequalities With Applications

The conventional enhancement-and-perturbation approach to establishing Gaussian extremal inequalities is refined via a novel monotone path argument in the product probability space. This refined approach is illustrated with simplified/corrected proofs of the Liu-Viswanath extremal inequality and a vector generalization of Costa's entropy power inequality. The power of this refinement is further demonstrated by characterizing two information-theoretic limits, namely, the capacity region of the multiple-input multiple-output (MIMO) Gaussian broadcast channel with private and common messages and the rate-distortion-equivocation function of vector Gaussian secure source coding, which have previously resisted the attack of the conventional approach.