Moderate Deviation Asymptotics for Variable-Length Codes With Feedback

Abstract

We consider data transmission across discrete memoryless channels (DMCs) using variable-length codes with feedback. We consider the family of such codes whose rates are $\rho _{N}$ below the channel capacity $C$ , where $\rho _{N}$ is a positive sequence that tends to zero slower than the reciprocal of the square root of the expectation of the (random) blocklength $N$ . This is known as the moderate deviations regime, and we establish the optimal moderate deviations constant. We show that in this scenario, the error probability decays sub-exponentially with speed $\exp (-(B/C)N\rho _{N})$ , where $B$ is the maximum relative entropy between output distributions of the DMC.