Redundancy of the Lempel-Ziv incremental parsing rule

The Lempel-Ziv codes are universal variable-to-fixed length codes that have become virtually standard in practical lossless data compression. For any given source output string from a Markov or unifilar source, we upper-bound the difference between the number of binary digits needed to encode the string and the self-information of the string. We use this result to demonstrate that for unifilar or Markov sources, the redundancy of encoding the first n letters of the source output with the Lempel-Ziv incremental parsing rule (LZ´78), the Welch modification (LZW), or a new variant is O((ln n)^-1), and we upper-bound the exact form of convergence. We conclude by considering the relationship between the code length and the empirical entropy associated with a string