Refined Asymptotics for the Fundamental Limits of Communications with Feedback

In this thesis, my contributions to communications with feedback in the context of information theory are divided into two parts:(1) establishing the fundamental limits for communications over the Gaussian MAC with fixed-and variable-length feedback codes and under the non-vanishing error probability formalism, and (2) establishing the moderate deviations asymptotics and joint-source channel coding exponent with variable-length feedback codes. In the first part, the ε-capacity regions for Gaussian MACs under expected power constraints with fixed-and variable-length codes with feedback are established. Prior to our works, all results pertaining to the non-vanishing error probability formalism are either restricted to channels with some form of symmetry, point-to-point channels, or channels with discrete (finite) alphabets. We invent new mathematical techniques based in part on information spectrum analyses and renewal theory (eg by Lai and Siegmund) to circumvent technical difficulties. For second part, exact values of the moderate deviation constant and the excess-distortion error exponent are established. Martingale-based techniques from Burnashev works as well as optional stopping theories are leveraged and analyzed in these works. Furthermore, a new decoding rule for joint-source channel coding—the MAP-distortion decoding rule—is developed and incorporated into our analysis of the excess-distortion error exponent. v