On the Convergence of the Polarization Process in the Noisiness/Weak-$\ast$ Topology

Let $W$ be a channel where the input alphabet is endowed with an Abelian group operation, and let $(W_n{)}_{n\ge 0}$ be Ar{\i}kan’s channel-valued polarization process that is obtained from $W$ using this operation. We prove that the process $(W_n{)}_{n\ge 0}$ converges almost surely to deterministic homomorphism channels in the noisiness/weak-$\ast$ topology. This provides a simple proof of multilevel polarization for a large family of channels, containing among others, discrete memoryless channels (DMC), and channels with continuous output alphabets. This also shows that any continuous channel functional converges almost surely (even if the functional does not induce a submartingale or a supermartingale).