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2020 On consecutive values of random completely multiplicative functions
Joseph Najnudel
Electron. J. Probab. 25(none): 1-28 (2020). DOI: 10.1214/20-EJP456

Abstract

In this article, we study the behavior of consecutive values of random completely multiplicative functions $(X_{n})_{n \geq 1}$ whose values are i.i.d. at primes. We prove that for $X_{2}$ uniform on the unit circle, or uniform on the set of roots of unity of a given order, and for fixed $k \geq 1$, $X_{n+1}, \dots , X_{n+k}$ are independent if $n$ is large enough. Moreover, with the same assumption, we prove the almost sure convergence of the empirical measure $N^{-1} \sum _{n=1}^{N} \delta _{(X_{n+1}, \dots , X_{n+k})}$ when $N$ goes to infinity, with an estimate of the rate of convergence. At the end of the paper, we also show that for any probability distribution on the unit circle followed by $X_{2}$, the empirical measure converges almost surely when $k=1$.

Citation

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Joseph Najnudel. "On consecutive values of random completely multiplicative functions." Electron. J. Probab. 25 1 - 28, 2020. https://doi.org/10.1214/20-EJP456

Information

Received: 9 May 2019; Accepted: 12 April 2020; Published: 2020
First available in Project Euclid: 5 May 2020

zbMATH: 07206398
MathSciNet: MR4095055
Digital Object Identifier: 10.1214/20-EJP456

Subjects:
Primary: 11K65, 11N37, 60F05, 60F15

JOURNAL ARTICLE
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