Electronic Journal of Probability

On the maximal length of arithmetic progressions

Minzhi Zhao and Huizeng Zhang

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This paper is a continuation of a paper by Benjamini, Yadin and Zeitouni's on maximal arithmetic progressions in random subsets. In this paper the asymptotic distributions of the maximal arithmetic progressions and arithmetic progressions modulo $n$ relative to an independent Bernoulli sequence with parameter $p$ are given. The errors are estimated by using the Chen-Stein method. Then the almost sure limit behaviour of these statistics is discussed. Our work extends the previous results and gives an affirmative answer to the conjecture raised at the end of that paper.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 79, 21 pp.

Accepted: 31 August 2013
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60C05: Combinatorial probability

arithmetic progression Bernoulli sequence limit distribution Chen-Stein method

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Zhao, Minzhi; Zhang, Huizeng. On the maximal length of arithmetic progressions. Electron. J. Probab. 18 (2013), paper no. 79, 21 pp. doi:10.1214/EJP.v18-2018. https://projecteuclid.org/euclid.ejp/1465064304

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