A Lower Bound on the Redundancy of Arithmetic-Type Delay Constrained Coding

In a previous paper we derived an upper bound on the redundancy of an arithmetic-type encoder for a memoryless source, designed to meet a finite end- to-end strict delay constraint. It was shown that the redundancy decays exponentially with the delay constraint and that the redundancy-delay exponent is lower bounded by log(1/alpha) where alpha is the probability of the most likely source symbol. In this work, we prove a corresponding upper bound for the redundancy-delay exponent, C - log 1/beta where beta is the probability of the least likely source symbol. This bound is valid for almost all memoryless sources and for all arithmetic-type (possibly time-varying, memory dependent) lossless delay-constrained encoders. We also shed some light on the difference between our exponential bounds and the polynomial O(d^-5´3) upper bound on the redundancy with an average delay constraint d, derived in an elegant paper by Bugeaud, Drmota and Szpankowski for another class of variable-to-variable encoders, and show that the difference is due to the precision needed to memorize the encoder´s state.