No input symbol should occur more frequently than 1-1/e

TL;DR

This paper proves that in any finite-input, finite-output discrete memoryless channel, the most frequent input symbol in a capacity-achieving distribution occurs less than 1-1/e times, and it explores conditions for capacity-achieving input distributions.

Contribution

It establishes an upper bound on the maximum probability of input symbols in capacity-achieving distributions and provides conditions for when a distribution can achieve capacity.

Findings

Maximum input symbol probability is less than 1-1/e in capacity-achieving distributions.

Provides sufficient conditions for a distribution to be capacity-achieving.

No similar restriction exists for output distributions.

Abstract

Consider any discrete memoryless channel (DMC) with arbitrarily but finite input and output alphabets X, Y respectively. Then, for any capacity achieving input distribution all symbols occur less frequently than 1-1/e$. That is, \[ \max\limits_{x \in \mathcal{X}} P^*(x) < 1-\frac{1}{e} \] \noindent where $P^*(x)$ is a capacity achieving input distribution. Also, we provide sufficient conditions for which a discrete distribution can be a capacity achieving input distribution for some DMC channel. Lastly, we show that there is no similar restriction on the capacity achieving output distribution.