Randomness Criterion Σ and Its Applications

Teturo Kamae (), Dong Han Kim () and Yu-Mei Xue ()
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Teturo Kamae: Osaka City University
Dong Han Kim: Dongguk University - Seoul
Yu-Mei Xue: BeiHang University

Sankhya A: The Indian Journal of Statistics, 2018, vol. 80, issue 2, No 9, 356-384

Abstract: 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.

Keywords: Randomness criterion; Champernowne number; Sturmian word; Primary 65C10; Secondary 11K45; 68R15 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s13171-017-0117-3

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