Second-Order Region for Gray-Wyner Network

TL;DR

This paper characterizes the second-order coding rates for the Gray-Wyner network using a tilted information density, providing a detailed analysis of the second-order region in a multi-terminal setting.

Contribution

It introduces a tilted information density and characterizes the second-order region for the Gray-Wyner network, a first for multi-terminal problems with auxiliary variables.

Findings

Second-order region characterized in terms of variance and tangent vector.

Achievability proved via type-covering argument.

Converse proved by refined perturbation approach.

Abstract

The coding problem over the Gray-Wyner network is studied from the second-order coding rates perspective. A tilted information density for this network is introduced in the spirit of Kostina-Verd\'u, and, under a certain regularity condition, the second-order region is characterized in terms of the variance of this tilted information density and the tangent vector of the first-order region. The second-order region is proved by the type method: the achievability part is proved by the type-covering argument, and the converse part is proved by a refinement of the perturbation approach that was used by Gu-Effros to show the strong converse of the Gray-Wyner network. This is the first instance that the second-order region is characterized for a multi-terminal problem where the characterization of the first-order region involves an auxiliary random variable.