POISSON GENERICITY IN NUMERATION SYSTEMS WITH EXPONENTIALLY MIXING PROBABILITIES

NICOLÁS ÁLVAREZ; VERÓNICA BECHER; EDA CESARATTO; MARTÍN MEREB; YUVAL PERES; BENJAMIN WEISS

doi:10.1090/tran/9592

POISSON GENERICITY IN NUMERATION SYSTEMS WITH EXPONENTIALLY MIXING PROBABILITIES

NICOLÁS ÁLVAREZ^*
, VERÓNICA BECHER
, EDA CESARATTO
, MARTÍN MEREB
, YUVAL PERES
, BENJAMIN WEISS

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We define Poisson genericity for infinite sequences in any countable alphabet with an invariant exponentially ψ -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.

Original language	English
Pages (from-to)	3403-3423
Number of pages	21
Journal	Transactions of the American Mathematical Society
Volume	379
Issue number	5
DOIs	https://doi.org/10.1090/tran/9592
State	Published - May 2026
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2025 by the authors

Keywords

Chen-Stein method
Numeration systems
Poisson processes
concentration inequalities

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title = "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 ψ -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.",

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AU - ÁLVAREZ, NICOLÁS

AU - BECHER, VERÓNICA

AU - CESARATTO, EDA

AU - MEREB, MARTÍN

AU - PERES, YUVAL

AU - WEISS, BENJAMIN

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N2 - We define Poisson genericity for infinite sequences in any countable alphabet with an invariant exponentially ψ -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.

AB - We define Poisson genericity for infinite sequences in any countable alphabet with an invariant exponentially ψ -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.

KW - Chen-Stein method

KW - Numeration systems

KW - Poisson processes

KW - concentration inequalities

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IS - 5

ER -

POISSON GENERICITY IN NUMERATION SYSTEMS WITH EXPONENTIALLY MIXING PROBABILITIES

Abstract

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Keywords

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