Polar Coding for Processes With Memory

We study polar coding for stochastic processes with memory. For example, a process may be defined by the joint distribution of the input and output of a channel. The memory may be present in the channel, the input, or both. We show that the <inline-formula> <tex-math notation="LaTeX">${\psi }$ </tex-math></inline-formula>-mixing processes polarize under the standard Arıkan transform, under a mild condition. We further show that the rate of polarization of the <italic>low-entropy</italic> synthetic channels is roughly <inline-formula> <tex-math notation="LaTeX">${O}({2}^{-\sqrt {N}})$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">${N}$ </tex-math></inline-formula> is the blocklength. That is, essentially, the same rate as in the memoryless case.