Almost sure optimality of sliding window Lempel-Ziv algorithm and variants

The sliding window Lempel Ziv algorithm has been studied extensively in the information theory literature and has been used in several commercial compression packages. In this paper, we establish the almost sure asymptotic optimality of the algorithm for the class of stationary and ergodic sources (without any mixing conditions) by using results on recurrence times from ergodic theory. We introduce a non-universal fixed shift version of the algorithm to gain more insight into its behavior. Asymptotic optimality results are also obtained for several variants introduced by Gavish and Lempel.