Global universality of the expected number of zeros of non-analytic random signals
Abstract
We study the asymptotics as n goes to infinity of 𝔼[𝒩(Sn,[0,2π])], the expected number of zeros in $[0, 2π]$ of a random periodic signal Sn of the form Sn(t)=∑k=1nak f(kt), where f is a non-analytic 2π−periodic function and the coefficients (ak) are i.i.d. random variables, centered with unit variance. We show in particular that if a1 admits a finite third moment and if the function f is piecewise polynomials and of class 𝒞7, then we have the following universal asymptotics, independent of the particular law of the coefficients (ak) limn → +∞Esp[𝒩(Sn,[0,2π])]/n= 2/√3√(|f'|L2([0,2π])/|f|L2([0,2π])). This result thus extends in expectation and at the scale of the whole period [0,2π] the local universality property established {in [Angst-Poly, IMRN, 2019]}, in distribution and in shrinking intervals of size 1/n. Moreover, it generalizes to a non-analytic context the global universality results obtained in the more classical frameworks of random trigonometric polynomials or random analytic functions. Our approach combines a new almost sure Central Limit Theorem à la Salem--Zygmund for the function Sn when evaluated at a uniform random point in $[0, 2π]$, and as well as suitable uniform integrability and anti-concentration estimates.
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BibTeX
@misc{angst2026global,
title = {{Global universality of the expected number of zeros of non-analytic random signals}},
author = {Jürgen Angst and Thibault Pautrel and Guillaume Poly},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01007},
eprint = {2609.01007},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01007},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global universality of the expected number of zeros of non-analytic random signals
AU - Jürgen Angst
AU - Thibault Pautrel
AU - Guillaume Poly
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01007
UR - https://arxiv.org/abs/2609.01007
AB - We study the asymptotics as n goes to infinity of 𝔼[𝒩(S_n,[0,2π])], the expected number of zeros in $[0, 2π]$ of a random periodic signal S_n of the form S_n(t)=∑_k=1^na_k f(kt), where f is a non-analytic 2π−periodic function and the coefficients (a_k) are i.i.d. random variables, centered with unit variance. We show in particular that if a_1 admits a finite third moment and if the function f is piecewise polynomials and of class 𝒞^7, then we have the following universal asymptotics, independent of the particular law of the coefficients (a_k) lim_n → +∞Esp[𝒩(S_n,[0,2π])]/n= 2/√3√(|f'|_L^2([0,2π])/|f|_L^2([0,2π])). This result thus extends in expectation and at the scale of the whole period [0,2π] the local universality property established {in [Angst-Poly, IMRN, 2019]}, in distribution and in shrinking intervals of size 1/n. Moreover, it generalizes to a non-analytic context the global universality results obtained in the more classical frameworks of random trigonometric polynomials or random analytic functions. Our approach combines a new almost sure Central Limit Theorem à la Salem--Zygmund for the function S_n when evaluated at a uniform random point in $[0, 2π]$, and as well as suitable uniform integrability and anti-concentration estimates.
ER -