A Proof of Non-stationary Channel Polarization

In this letter, we consider the proof of non-stationary channel polarization theory. First we construct a multi-channel stochastic process for the non-stationary channel polarization operation. Then based on this stochastic process, we extend Ar{\i}kan's standard martingale proof method on the average channel capacity and average channel Bhattacharyya parameter, by which we have proved the non-stationary channel polarization theory.