AI 中文总结
该研究证明了非解析随机周期信号零点期望个数的全局普适性，扩展了相关局部普适性结果，采用新的中心极限定理及一致可积性等估计完成推导。
AI 中文摘要
我们研究当n趋向无穷时，随机周期信号S_n在区间[0,2π]上零点的期望个数E[N(S_n,[0,2π])]的渐近行为，其中S_n(t)的形式为S_n(t)=∑_{k=1}^{n}a_k f(kt)，f是非解析的2π周期函数，系数(a_k)是独立同分布的随机变量，均值为0，方差为1。我们特别证明，若a_1具有有限的三阶矩，且f是分段多项式且属于C^7类，则存在与系数(a_k)的具体分布无关的普适渐近关系：当n趋向正无穷时，lim [E[N(S_n,[0,2π])]/n] = (2/√3)√(||f'||_{L^2([0,2π])}/||f||_{L^2([0,2π])})。该结果在期望层面和整个周期[0,2π]的尺度上，扩展了[Angst-Poly, IMRN, 2019]中在分布层面和大小为1/n的收缩区间上建立的局部普适性，还将更经典的随机三角多项式或随机解析函数框架下的全局普适性结果推广到了非解析情形。我们的方法结合了一种新的、类似Salem-Zygmund的几乎处处中心极限定理（用于S_n在[0,2π]内均匀随机点处的取值），以及合适的一致可积性和反集中估计。
英文摘要
We study the asymptotics as $n$ goes to infinity of $\mathbb E\left[\mathcal{N}(S_n,[0,2π])\right]$, the expected number of zeros in $[0, 2π]$ 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 $2π-$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 $\mathcal C^{7}$, then we have the following universal asymptotics, independent of the particular law of the coefficients $(a_k)$ \[ \lim_{n \to +\infty}\frac{\Esp\left[\mathcal{N}(S_n,[0,2π])\right]}{n}= \frac{2}{\sqrt{3}}\sqrt{\frac{\|f'\|_{L^2([0,2π])}}{\|f\|_{L^2([0,2π])}}}. \] 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.
