On the coding rate of a multishot Tunstall code for stationary memoryless sources

The coding rate of a multishot Tunstall code for stationary memoryless sources is investigated in non-universal situation so that the probability distribution of a source is known to the encoder and the decoder. Tunstall code is the original and an optimal variable-to-fixed length (VF) source code. A multishot VF code parses a given source sequence to variable-length blocks and encodes them to fixed-length codewords. By concatenating the codewords, the codeword for the concatenation of all the parsed blocks is obtained. When studying one-shot or multishot VF codes, the average coding rate has been defined as the codeword length divided by the average block length. In this paper, the average coding rate is defined as the expectation of the pointwise coding rate for the sequences which are parsed to the same number of blocks. It is proved that the average coding rate defined in this manner converges to the entropy of the stationary memoryless source under the assumption that the geometric mean of the leaf counts of the multishot Tunstall parsing trees goes to infinity. The result is a straightforward extension of the analysis of the one-shot Tunstall code by the author. In the proof, Cartesian concatenation of Tunstall trees and the geometric mean of the leaf counts of Tunstall trees play crucial roles.