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 robust-ness of this property under small violations of ergodicity is studied. A notion of deficiency of algorithmic randomness is used as a measure of disagreement between data sequence and probability measure. We prove that universal compressing schemes from a large class are non-robust 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 compressing algorithm can be violated. Lempel–Ziv compressing algorithms are robust on infinite sequences generated by ergodic Markov chains when the randomness deficiency of its initial fragments of length n grows as o(n).