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.

中文翻译：

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关于非解析随机周期信号的零点

在本文中，我们研究了 $S_n(t)=\sum_{k=1}^n a_k f(kt)$ 形式的随机周期信号的零点数的局部普遍性，其中 $f$ 是一个$2\pi-$periodic 函数满足弱正则性条件，其中系数 $a_k$ 是 iid 随机变量，以单位方差为中心。特别是，我们的结果适用于连续分段线性函数。我们证明 $S_n(t)$ 在大小为 $1/n$ 的收缩区间中的零点数在法则上收敛，因为 $n$ 趋于无穷大到高斯过程的零点数，其显式协方差仅取决于函数 $f$ 而不是基于随机系数 $(a_k)$ 的普通定律。作为副产品，这意味着 $S_n(t)$ 零点的点测度在法律上收敛到 $\mathbb R$ 上的局部有限点测度空间的明确限制，赋予模糊拓扑。用于建立这种普遍性结果的涉及 $f$ 的规律性甚至分析性的标准工具在这里被最近在 [CCK17] 中获得的一些高维 Berry-Esseen 边界所取代。后者允许我们在由于缺乏规律性而无法应用通常的标准的情况下证明 $C^1$ 拓扑中的功能 CLT。