Zeros of smooth stationary Gaussian processes

Abstract

Let $f:\mathbb{R}\to \mathbb{R}$ be a stationary centered Gaussian process. For any $R>0$, let ${\nu }_R$ denote the counting measure of $\{x\in \mathbb{R}\mid f(Rx)=0\}$. In this paper, we study the large $R$ asymptotic distribution of ${\nu }_R$. Under suitable assumptions on the regularity of $f$ and the decay of its correlation function at infinity, we derive the asymptotics as $R\to +\infty$ of the central moments of the linear statistics of ${\nu }_R$. In particular, we derive an asymptotics of order $R^{\frac{p}{2}}$ for the $p$-th central moment of the number of zeros of $f$ in $[0,R]$. As an application, we derive a functional Law of Large Numbers and a functional Central Limit Theorem for the random measures~${\nu }_R$. More precisely, after a proper rescaling, ${\nu }_R$ converges almost surely towards the Lebesgue measure in weak-$\ast$ sense. Moreover, the fluctuation of ${\nu }_R$ around its mean converges in distribution towards the standard Gaussian White Noise. The proof of our moments estimates relies on a careful study of the $k$-point function of the zero point process of~$f$, for any $k\ge 2$. Our analysis yields two results of independent interest. First, we derive an equivalent of this $k$-point function near any point of the large diagonal in~${\mathbb{R}}^k$, thus quantifying the short-range repulsion between zeros of $f$. Second, we prove a clustering property which quantifies the long-range decorrelation between zeros of $f$.

References in corpus (5)

• Distribution of zeros of random and quantum chaotic sections of positive line bundles
• Universality and scaling of correlations between zeros on complex manifolds
• Level crossings and other level functionals of stationary Gaussian processes
• Limit theory for geometric statistics of point processes having fast decay of correlations
• Variance of the volume of random real algebraic submanifolds II

Cited by in corpus (1)

• Overcrowding estimates for zero count and nodal length of stationary Gaussian processes