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-$*$ 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 $k \geq 2$. 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 $f$.
Highlights
Let Z denote the zero set of a smooth centered stationary Gaussian process f on R
We prove that ρk satisfies a clustering property if the correlation function of the process f decays fast enough
It is paramount in their work that the Bargmann–Fock is the restriction to R of a Gaussian Entire Function. In both [30] and [18], the authors deduce from their clustering result a Central Limit Theorem for the linear statistics of the point processes they study
Summary
Let Z denote the zero set of a smooth centered stationary Gaussian process f on R. A classical problem in probability is to understand the number of zeros of f in a growing interval, that is the asymptotics of Card(Z ∩ [0, R]) as R → +∞. This problem has a long history, starting with the articles of Kac [23] and Rice [33] who computed the mean number of zeros of f in an interval. We prove that ρk satisfies a clustering property if the correlation function of the process f decays fast enough. This clustering property can be interpreted as a clue that zeros of f in two disjoint intervals that are far from one another are quasi-independent. We believe that the methods we develop below regarding these divided differences can have applications beyond the scope of this paper
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have