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We investigate, starting with a symmetric B-DMC, the evolution of various probabilities related to the likelihood ratios of the synthetic channels created by the recursive application of the polarization transformations. We show the processes are bounded sub/super martingales converging to the extremes of the bounded intervals. In particular, this is true for the process that tracks the error probability of the likelihood ratio based decision rule of the synthetic channels. Moreover, the derivations reveal the set of BECs do not only provide universally good indices for the information sets of symmetric B-DMCs, but also universally bad indices. The analysis provides a new perspective into the theory of channel polarization initiated by Arikan, and it helps us to address an approximation to the computations of the likelihood ratios of the synthetic channels.

Properties of the polarization transformations for the likelihood ratios of symmetric B-DMCs