Poisson genericity in numeration systems with exponentially mixing probabilities

TL;DR

This paper introduces the concept of Poisson genericity for sequences in various alphabets with exponentially mixing measures, proving that almost all such sequences exhibit Poisson-distributed block occurrences, generalizing previous results.

Contribution

It generalizes the notion of Poisson genericity to broader numeration systems with exponentially mixing probabilities, showing that almost all sequences are Poisson generic.

Findings

Almost all sequences are Poisson generic.

Generalization of previous Poisson genericity results.

Continued fraction expansions are Poisson generic for almost all real numbers.

Abstract

We define Poisson genericity for infinite sequences in any finite or countable alphabet with an invariant exponentially-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 integral bases numeration systems. In particular, we obtain that their continued fraction expansions for almost all real numbers are Poisson generic.