Moderate Deviation Asymptotics for Variable-Length Codes With Feedback

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.