On colorings of bivariate random sequences
František Matúš, Michal Kupsa · 2010
The ergodic sequences consisting of vectors (ξn, ηn), n ≥ 1, over a finite alphabet A×B are colored with ⌊enα⌋ colors for Anand⌊enβ⌋ colors for Bn. Generic behavior of the colorings in terms of probabilities of monochromatic rectangles intersected with typical sets is examined. When n increases a big majority of pairs of colorings produces rectangles whose probabilities are bounded uniformly from above. Limiting rates of bounds are worked out in all regimes of the rates α and β of colorings. As a consequence, generic behavior of the colorings in terms of Shannon entropies of the partitions into rectangles is described.