  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.
