On the Capacity of Deletion Channels with States

We consider the class of channels formed from the concatenation of a deletion channel and a finite-state channel. For this class of channels, we show that the operationally-defined capacity is equal to the stationary capacity. We also show that the stationary capacity can be approached by a sequence of Markov processes with increasing Markovian orders. As a byproduct, we show that the polar coding scheme constructed by Tal, Pfister, Fazeli, and Vardy [arxiv: 1904.13385 (2019)] achieves the capacity of the binary deletion channel.