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 ${\\psi }$ -mixing processes polarize under the standard Arikan transform, under a mild condition. We further show that the rate of polarization of the low-entropy synthetic channels is roughly ${O}({2}^{-\\sqrt {N}})$ , where ${N}$ is the blocklength. That is, essentially, the same rate as in the memoryless case.