Second-Order Asymptotics for the Scalar Gaussian Broadcast Channel

Abstract

This paper characterizes the second-order coding region of the scalar Gaussian broadcast channel along the curved boundary of its capacity region. The characterization holds under both average and maximal error criteria and allows correlation between the receiver noises. At each interior point of this boundary, the second-order region is a halfspace, as in the work of Scarlett and the author on Gaussian multiple-access channels with degraded message sets. Achievability follows from spherical superposition coding with a blocklength-dependent power allocation. The main innovation lies in the converse. Using Ornstein--Uhlenbeck entropy dissipation and Gaussian functional inequalities, we construct an auxiliary observation through an additive Gaussian noise channel and establish quantitative relative entropy bounds for the channel output distributions conditioned on this observation. The strong functional representation lemma of Li and El Gamal then allows us to select a realization of its independent random seed to obtain a deterministic partition of the retained receiver-1 message set. The component label plays a role analogous to the auxiliary random variable in the capacity characterization, while providing the distributional control needed at the second-order scale. These properties guide the construction of auxiliary output measures whose likelihood ratios admit the cancellations and normal approximation needed to evaluate an information spectrum converse.