论文标题
光滑的固定高斯过程的零
Zeros of smooth stationary Gaussian processes
论文作者
论文摘要
令$ f：\ mathbb {r} \ to \ mathbb {r} $为固定的居中高斯进程。对于任何$ r> 0 $，令$ν_r$表示$ \ {x \ in \ Mathbb {r} \ mid f（rx）= 0 \} $的计数度量。在本文中，我们研究了$ν_r$的大$ r $渐近分布。在适当的假设$ f $及其相关函数的衰减下，我们将渐近技术定为$ r \ to +\ to +\ infty $的$ν_r$的线性统计信息的中心时刻。特别是，我们得出了$ r^\ frac {p} {2} $的顺序渐近学，用于$ p $ - $ f $ in $ [0，r] $的零零数的中心矩。作为一个应用程序，我们得出了大量的功能定律和随机度量〜$ν_r$的功能性中心限制定理。更确切地说，经过适当的重新进行重新进行后，$ν_r$几乎肯定会以弱的 - $*$ Sense收敛。此外，$ν_r$围绕其平均值的波动将分布的分布收敛于标准高斯白噪声。我们时刻的证明估计依靠对任何$ K \ geq 2 $的零点过程的$ K $点功能的仔细研究。我们的分析产生了两个独立关注的结果。首先，我们在〜$ \ mathbb {r}^k $中的任何点的任何点中得出了相当于此$ k $ - 点功能，从而量化了$ f $的零之间的短距离排斥。其次，我们证明了一个聚类属性，该属性量化了$ f $的零之间的远程去相关。
Let $f:\mathbb{R} \to \mathbb{R}$ be a stationary centered Gaussian process. For any $R>0$, let $ν_R$ denote the counting measure of $\{x \in \mathbb{R} \mid f(Rx)=0\}$. In this paper, we study the large $R$ asymptotic distribution of $ν_R$. 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 $ν_R$. 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~$ν_R$. More precisely, after a proper rescaling, $ν_R$ converges almost surely towards the Lebesgue measure in weak-$*$ sense. Moreover, the fluctuation of $ν_R$ 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 $f$.
