  A transform that is universally polarizing over a set of channels with memory
is presented. Memory may be present in both the input to the channel and the
channel itself. Both the encoder and the decoder are aware of the input
distribution, which is fixed. However, only the decoder is aware of the actual
channel being used. The transform can be used to design a universal code for
this scenario. The code is to have vanishing error probability when used over
any channel in the set, and achieve the infimal information rate over the set.
The setting considered is, in fact, more general: we consider a set of
processes with memory. Universal polarization is established for the case where
each process in the set: (a) has memory in the form of an underlying hidden
Markov state sequence that is aperiodic and irreducible, and (b) satisfies a
`forgetfulness' property. Forgetfulness, which we believe to be of independent
interest, occurs when two hidden Markov states become approximately independent
of each other given a sufficiently long sequence of observations between them.
We show that aperiodicity and irreducibility of the underlying Markov chain is
not sufficient for forgetfulness, and develop a sufficient condition for a
hidden Markov process to be forgetful.
