We explore the fundamental limits for distributed lossless source coding
(Slepian-Wolf) in the ﬁnite blocklength regime. We introduce a fundamental
quantity known
as the entropy dispersion matrix, which is analogous to scalar dispersion
quantities. If this dispersion matrix is positive-deﬁnite, the optimal rate
region under the constraint of a ﬁxed blocklength and non-zero error
probability has a curved boundary compared to being polyhedral in the asymptotic
case. Furthermore, we make use of the multidimensional Berry-Esseen theorem to
show that the dispersion matrix exactly characterizes the rate of convergence of
the non-asymptotic rate region to the asymptotic one.