Poisson genericity in numeration systems with exponentially mixing probabilities

Abstract

We define Poisson genericity for infinite sequences in any countable alphabet with an invariant exponentially phi-mixing probability measure. A sequence is Poisson generic if the number of occurrences of blocks of symbols asymptotically follows a Poisson law as the block length increases. We prove that almost all sequences are Poisson generic. Our result generalizes Peres and Weiss’ theorem about Poisson genericity of integer bases numeration systems. In particular, we obtain that the continued fraction expansions of almost all real numbers are Poisson generic