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

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