  Our understanding of information in systems has been based on the foundation
of memoryless processes. Extensions to stable Markov and auto-regressive
processes are classical. Berger proved a source coding theorem for the
marginally unstable Wiener process, but the infinite-horizon exponentially
unstable case has been open since Gray's 1970 paper. There were also no
theorems showing what is needed to communicate such processes across noisy
channels.
  In this work, we give a fixed-rate source-coding theorem for the
infinite-horizon problem of coding an exponentially unstable Markov process.
The encoding naturally results in two distinct bitstreams that have
qualitatively different QoS requirements for communicating over a noisy medium.
The first stream captures the information that is accumulating within the
nonstationary process and requires sufficient anytime reliability from the
channel used to communicate the process. The second stream captures the
historical information that dissipates within the process and is essentially
classical. This historical information can also be identified with a natural
stable counterpart to the unstable process. A converse demonstrating the
fundamentally layered nature of unstable sources is given by means of
information-embedding ideas.
