A Tight Upper Bound for the Third-Order Asymptotics for Most Discrete Memoryless Channels

TL;DR

This paper establishes a precise third-order upper bound on the epsilon-error capacity for discrete memoryless channels, matching known lower bounds and clarifying the asymptotic behavior depending on the channel's dispersion.

Contribution

It provides a tight third-order upper bound for the epsilon-error capacity of discrete memoryless channels, extending previous second-order results.

Findings

Upper bound matches the lower bound for channels with positive dispersion.

For channels with vanishing dispersion, the capacity is tightly bounded by n times the capacity plus a constant.

The third-order term does not exceed 1/2 log n + O(1) for channels with positive epsilon-dispersion.

Abstract

This paper shows that the logarithm of the epsilon-error capacity (average error probability) for n uses of a discrete memoryless channel is upper bounded by the normal approximation plus a third-order term that does not exceed 1/2 log n + O(1) if the epsilon-dispersion of the channel is positive. This matches a lower bound by Y. Polyanskiy (2010) for discrete memoryless channels with positive reverse dispersion. If the epsilon-dispersion vanishes, the logarithm of the epsilon-error capacity is upper bounded by the n times the capacity plus a constant term except for a small class of DMCs and epsilon >= 1/2.