Randomness Criterion Σ and Its Applications

Kamae, Teturo; Kim, Dong Han; Xue, Yumei

doi:10.1007/s13171-017-0117-3

Cited by 3 publications

(2 citation statements)

References 8 publications

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“…Also, Peres and Weiss proved that Poisson genericity implies Borel normality and that the two notions are not equivalent, witnessed by the fact that Champernowne's sequence is not Poisson generic. Weiss described their proof in [13] and the method was transcribed by Kamae et al [9,8] applied to other randomness notions.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Poisson generic sequences

Álvarez¹,

Becher²,

Mereb³

2022

Preprint

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Years ago, Zeev Rudnick defined Poisson generic sequences as those where the number of occurrences of the long strings in the initial segments of the sequences follow the Poisson distribution. Peres and Weiss proved that almost all sequences, with respect to Lebesgue measure, are Poisson generic, but they did not publish their proof. In this note first we transcribe Peres and Weiss' proof and then we show that there are computable Poisson generic instances and that all Martin-Löf random sequences are Poisson generic.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Poisson generic sequences

Álvarez¹,

Becher²,

Mereb³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Theorem 2 is a generalization of the result proved by Weiss that 1-Poisson genericity implies normality. Weiss' result is reported by Kamae et al in [10] and the proof technique is used in [9,Theorem 2] to show that the related notion of P -randomness implies normality. We essentially follow the proof outlined by Weiss in [16].…”

Section: Proof Of Theoremmentioning

confidence: 99%

A construction of a $λ$- Poisson generic sequence

Becher¹,

Himelfarb²

2022

Preprint

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Years ago Zeev Rudnick defined the λ-Poisson generic sequences as the infinite sequences of symbols in a finite alphabet where the number of occurrences of long words in the initial segments follow the Poisson distribution with parameter λ. Although it has long been known that almost all sequences, with respect to Lebesgue measure, are Poisson generic, no explicit instance has yet been given. In this note we give a construction of an explicit λ-Poisson generic sequence over an alphabet of at least three symbols, for any fixed positive real number λ. Since λ-Poisson genericity implies Borel normality, the constructed sequence is Borel normal. The same construction provides explicit instances of Borel normal sequences that are not λ-Poisson generic.

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Poisson Generic Sequences

Álvarez

Becher

Mereb

2022

International Mathematics Research Notices

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Years ago, Zeev Rudnick defined the Poisson generic real numbers by counting the number of occurrences of long blocks of digits in the initial segments of the expansions of the real numbers in a fixed integer base. Peres and Weiss proved that almost all real numbers, with respect to the Lebesgue measure, are Poisson generic, but they did not publish their proof. In this note, we first transcribe Peres and Weiss’ proof and then we show that there are computable Poisson generic instances and that all Martin–Löf random real numbers are Poisson generic.

show abstract

Randomness Criterion Σ and Its Applications

Cited by 3 publications

References 8 publications

Poisson generic sequences

Poisson generic sequences

A construction of a $λ$- Poisson generic sequence

Poisson Generic Sequences

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