On the Rate of Channel Polarization

It is shown that for any binary-input discrete memoryless channel $W$ with symmetric capacity $I(W)$ and any rate $R <I(W)$, the probability of block decoding error for polar coding under successive cancellation decoding satisfies $P_e \le 2^{-N\beta}$ for any $\beta<\frac12$ when the block-length $N$ is large enough.