Second-Order Asymptotics for the Gaussian MAC on the Relative Interior of the Sum-Rate Face

RESUMO

We completely characterize the second-order coding rate region of the two-user Gaussian multiple-access channel at points in the relative interior of the sum-rate face, under maximal per-codeword power constraints. For any fixed average error probability $0 < \varepsilon < 1/2$, a nontrivial mixture of power splits reduces the sum-rate dispersion and strictly improves upon standard inner bounds based on a constant power split. Thus, a constant power split, although sufficient to attain the first-order capacity region, is insufficient for second-order optimality in this regime. Our achievability proof builds on the constant composition and coded time-sharing framework of Scarlett, Martinez, and Guillén i Fàbregas, with an optimization over admissible power splits. A matching second-order converse establishes the optimality of the resulting coded time-sharing strategy.