Asymptotic capacity of a random channel

Abstract

We consider discrete memoryless channels with input and output alphabet size n whose channel transition matrix consists of entries that are independent and identically distributed according to some probability distribution v on (R≥0, B(R≥0)) before being normalized, where v is such that E[X log X)² 1 <; ∞, μ₁ := E[X] and μ₂ := E[X log X] for a random variable X with distribution v. We prove that in the limit as n → ∞, the capacity of such a channel converges to μ₂/μ₁ - log μ₁ almost surely and in L². We further show that the capacity of these random channels converges to this asymptotic value exponentially in n.