Redundancy of the Lempel-Ziv-Welch code

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 of unifilar source, we upper bound the difference between the number of binary digits needed by the Lempel-Ziv-Welch code (1977, 1978, 1984) to encode the string and the self-information of the string. We use this result to demonstrate that for unifilar, Markov sources, the redundancy of encoding the first n letters of the source output with LZW is O((ln n)^-1), and we upper bound the exact form of convergence