Almost sure convergence of the L4 norm of Littlewood polynomials

Yongjiang Duan, Xiang Fang, Na Zhan*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

This paper concerns the norm of Littlewood polynomials on the unit circle which are given by (formula presented) i.e., they have random coefficients in {-1,1}. Let (formula presented) We show that (formula presented) almost surely as n → ∞. This improves a result of Borwein and Lockhart (2001, Proceedings of the American Mathematical Society 129, 1463-1472), who proved the corresponding convergence in probability. Computer-generated numerical evidence for the a.s. convergence has been provided by Robinson (1997, Polynomials with plus or minus one coefficients: growth properties on the unit circle, M.Sc. thesis, Simon Fraser University). We indeed present two proofs of the main result. The second proof extends to cases where we only need to assume a fourth moment condition.

Original languageEnglish
Pages (from-to)872-885
Number of pages14
JournalCanadian Mathematical Bulletin
Volume67
Issue number3
DOIs
StatePublished - 1 Sep 2024

Keywords

  • almost sure convergence
  • L norm
  • Littlewood polynomial
  • Serfling's maximal inequality

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