Second-Order Asymptotics for the Gaussian Multiple-Access Channel at Corner Points

Abstract

We establish exact second-order coding rate regions at the two corner points of the capacity region of the two-user Gaussian multiple-access channel. For any average error probability $\epsilon \in (0,1)$, we characterize the $n^{-1/2}$-scale fluctuations of achievable rates around each corner point, proving a converse that matches the previously known achievability bound. The proof first extracts a rectangular subcode that preserves the independence of the two transmitted codewords. The Polyanskiy--Verdú good-code output distribution theorem then yields log-determinant constraints that induce a spectral decomposition of a trimmed codebook into diffuse and exceptional subspaces. An entropic Brascamp--Lieb projection inequality controls the codeword inner product on the diffuse subspace, while variance-inflated Gaussian output distributions handle the low-dimensional exceptional subspace. Combining these two treatments yields the joint Gaussian limit required for the matching second-order converse.

28 pages, 4 figures