Problems of Robustness for Universal Coding Schemes

The Lempel–Ziv universal coding scheme is asymptotically optimal for the class of all stationary ergodic sources. A problem of robustness of this property under small violations of ergodicity is studied. The notion of deficiency of algorithmic randomness is used as a measure of disagreement between data sequence and probability measure. We prove that universal compression schemes from a large class are nonrobust in the following sense: If the randomness deficiency grows arbitrarily slowly on initial fragments of an infinite sequence, then the property of asymptotic optimality of any universal compression algorithm can be violated. Lempel–Ziv compression algorithms are robust on infinite sequences generated by ergodic Markov chains when the randomness deficiency of their initial fragments of length $n$ grows as $o(n)$.