A simple technique for bounding the pointwise redundancy of the 1978 Lempel-Ziv algorithm

If x is a string of finite length over a finite alphabet A, let LZ(x) denote the length of the binary codeword assigned to x by the 1978 version of the Lempel-Ziv data compression algorithm, let t(x) be the number of phrases in the Lempel-Ziv parsing of x, and let μ(x) be the probability assigned to x by a memoryless source model. Using a very simple technique, we probe the pointwise redundancy bound LZ(x)+log₂μ(x)⩽8t(x)max{-log₂ μ(a):a∈A}