Capacity of Random Channels with Large Alphabets
Journal
Advances in Mathematics of Communications, 2017
Type
Controlled Vocabulary for Resource Type Genres::other
Date Issued
2015-03-13
Author(s)
Abstract
We consider discrete memoryless channels with input alphabet size $n$ and output alphabet size $m$, where $m=$ceil$(\gamma n)$ for some constant $\gamma > 0$. The channel transition matrix consists of entries that, before being normalised, are independent and identically distributed nonnegative random variables $V$ and such that $E[(V \log V)^2 ] < \infty$. We prove that in the limit as $n\to \infty$ the capacity of such a channel converges to $Ent(V) / E[V ] $ almost surely and in $L^2$, where $Ent(V):= E[V\log V ] -E[V ] \log E[V ] $ denotes the entropy of $V$. We further show that, under slightly different model assumptions, the capacity of these random channels converges to this asymptotic value exponentially in $n$. Finally, we present an application in the context of Bayesian optimal experiment design.