Moderate Deviation Asymptotics for Variable-Length Codes with Feedback

TL;DR

This paper analyzes the decay rate of error probabilities in variable-length feedback codes over DMCs when the rate approaches capacity at a specific moderate deviations scale, establishing the optimal decay constant.

Contribution

It derives the optimal moderate deviations constant for variable-length feedback codes in the regime where the rate approaches capacity slowly.

Findings

Error probability decays sub-exponentially with rate \\exp(-(B/C)N\\rho_N)

Established the optimal moderate deviations constant for the regime

Characterized the decay speed in the moderate deviations setting

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.