“Any-time” capacity and a separation theorem for tracking unstable processes

The problem of tracking an exponentially unstable scaler source process across a noisy channel is considered. We introduce the a new parametric notion of capacity that we call “any-time capacity” C_at(α). It is a twist on the familiar concept of error-exponents and is always between the classical Shannon capacity and the zero-error capacity. A separation theorem is given which shows that C_at(α) characterizes the properties of a channel needed for finite expected distortion