Global universality of the expected number of zeros of non-analytic random signals

We study the asymptotics as $n$ goes to infinity of $\mathbb{E}[\mathcal{N}(S_n,[0,2\pi ])]$, the expected number of zeros in $[0,2\pi ]$ of a random periodic signal $S_n$ of the form $$S_n(t)=\sum _{k=1}^n{a}_{k}f(kt),$$ where $f$ is a non-analytic $2\pi -$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 ${\mathcal{C}}^7$, then we have the following universal asymptotics, independent of the particular law of the coefficients $(a_k)$ $$\underset{n\to +\mathrm{\infty }}{\lim }\frac{\text{Esp}[\mathcal{N}(S_n,[0,2\pi ])]}{n}=\frac{2}{\sqrt{3}}\sqrt{\frac{\mathrm{\parallel }{f}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2([0,2\pi ])}}{\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2([0,2\pi ])}}}\mathrm{.}$$ This result thus extends in expectation and at the scale of the whole period $[0,2\pi ]$ the local universality property established {in [Angst-Poly, IMRN, 2019]}, in distribution and in shrinking intervals of size $1\mathrm{/}n$. 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 $[0,2\pi ]$, and as well as suitable uniform integrability and anti-concentration estimates.