  This paper studies the second-order asymptotics of the Gaussian
multiple-access channel with degraded message sets. For a fixed average error
probability $\varepsilon \in (0,1)$ and an arbitrary point on the boundary of
the capacity region, we characterize the speed of convergence of rate pairs
that converge to that boundary point for codes that have asymptotic error
probability no larger than $\varepsilon$. As a stepping stone to this local
notion of second-order asymptotics, we study a global notion, and establish
relationships between the two. We provide a numerical example to illustrate how
the angle of approach to a boundary point affects the second-order coding rate.
This is the first conclusive characterization of the second-order asymptotics
of a network information theory problem in which the capacity region is not a
polygon.
