On consecutive values of random completely multiplicative functions

Joseph Najnudel*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

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Abstract

In this article, we study the behavior of consecutive values of random completely multiplicative functions (Xn)n≥1 whose values are i.i.d. at primes. We prove that for X2 uniform on the unit circle, or uniform on the set of roots of unity of a given order, and for fixed k ≥ 1, Xn+1,...,Xn+k are independent if n is large enough. Moreover, with the same assumption, we prove the almost sure convergence of the empirical measure N−1PN n=1 δ(Xn+1,...,Xn+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 X2, the empirical measure converges almost surely when k = 1.
Original languageEnglish
Article number59
Number of pages28
JournalElectronic Journal of Probability
Volume25
DOIs
Publication statusPublished - 2020

Keywords

  • random multiplicative function
  • empirical distribution
  • limit theorem
  • Chowla conjecture

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