Randomness Criterion Σ and Its Applications

Teturo Kamae; Dong Han Kim; Yu Mei Xue

doi:10.1007/s13171-017-0117-3

Randomness Criterion Σ and Its Applications

Teturo Kamae
, Dong Han Kim
, Yu Mei Xue

Department of Mathematics Education

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

1 Scopus citations

Abstract

The Sigma function, which is the sum of the squares of the number of occurrences of every factor, is a criterion of randomness, measuring specially the uniformity of the block distribution. An infinite word whose prefixes attain asymptotically the smallest possible value of it is called Sigma-random. We prove that the Champernowne word is Sigma-random. We also consider less complex words which have values with asymptotically larger order, Sturmian words and almost 0-words.

Original language	English
Pages (from-to)	356-384
Number of pages	29
Journal	Sankhya A
Volume	80
Issue number	2
DOIs	https://doi.org/10.1007/s13171-017-0117-3
State	Published - 1 Aug 2018

Keywords

68R15
Champernowne number
Primary 65C10
Randomness criterion
Secondary 11K45
Sturmian word

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10.1007/s13171-017-0117-3

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abstract = "The Sigma function, which is the sum of the squares of the number of occurrences of every factor, is a criterion of randomness, measuring specially the uniformity of the block distribution. An infinite word whose prefixes attain asymptotically the smallest possible value of it is called Sigma-random. We prove that the Champernowne word is Sigma-random. We also consider less complex words which have values with asymptotically larger order, Sturmian words and almost 0-words.",

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author = "Teturo Kamae and Kim, \{Dong Han\} and Xue, \{Yu Mei\}",

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N2 - The Sigma function, which is the sum of the squares of the number of occurrences of every factor, is a criterion of randomness, measuring specially the uniformity of the block distribution. An infinite word whose prefixes attain asymptotically the smallest possible value of it is called Sigma-random. We prove that the Champernowne word is Sigma-random. We also consider less complex words which have values with asymptotically larger order, Sturmian words and almost 0-words.

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KW - Champernowne number

KW - Primary 65C10

KW - Randomness criterion

KW - Secondary 11K45

KW - Sturmian word

UR - https://www.scopus.com/pages/publications/85036588739

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JO - Sankhya A

JF - Sankhya A

IS - 2

ER -

Randomness Criterion Σ and Its Applications

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Keywords

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