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Article Dans Une Revue International Mathematics Research Notices Année : 2022

On the zeros of non-analytic random periodic signals

Résumé

In this paper, we investigate the local universality of the number of zeros of a random periodic signal of the form $S_n(t)=\sum_{k=1}^n a_k f(k t)$, where $f$ is a $2\pi-$periodic function satisfying weak regularity conditions and where the coefficients $a_k$ are i.i.d. random variables, that are centered with unit variance. In particular, our results hold for continuous piecewise linear functions. We prove that the number of zeros of $S_n(t)$ in a shrinking interval of size $1/n$ converges in law as $n$ goes to infinity to the number of zeros of a Gaussian process whose explicit covariance only depends on the function $f$ and not on the common law of the random coefficients $(a_k)$. As a byproduct, this entails that the point measure of the zeros of $S_n(t)$ converges in law to an explicit limit on the space of locally finite point measures on $\mathbb R$ endowed with the vague topology. The standard tools involving the regularity or even the analyticity of $f$ to establish such kind of universality results are here replaced by some high-dimensional Berry-Esseen bounds recently obtained in [CCK17]. The latter allow us to prove functional CLT's in $C^1$ topology in situations where usual criteria can not be applied due to the lack of regularity.

Dates et versions

hal-02359402 , version 1 (12-11-2019)

Identifiants

Citer

Jürgen Angst, Guillaume Poly. On the zeros of non-analytic random periodic signals. International Mathematics Research Notices, 2022, 7, pp.4931-4968. ⟨10.1093/imrn/rnaa201⟩. ⟨hal-02359402⟩
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