On the Zeros of Non-Analytic Random Periodic Signals

Abstract

Abstract 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, which 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 [ 7]. The latter allow us to prove functional Central Limit Theorems in $C^1$ or Lipschitz topology in situations where usual criteria cannot be applied due to the lack of regularity.