Electronic Journal of Probability

On the maximal length of arithmetic progressions

Minzhi Zhao and Huizeng Zhang

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465064304

Digital Object Identifier
doi:10.1214/EJP.v18-2018

Mathematical Reviews number (MathSciNet)
MR3101645

Zentralblatt MATH identifier
1285.60022

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

Keywords
arithmetic progression Bernoulli sequence limit distribution Chen-Stein method

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

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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