• Bernoulli
  • Volume 24, Number 2 (2018), 1128-1170.

Critical points of multidimensional random Fourier series: Central limits

Liviu I. Nicolaescu

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We investigate certain families $X^{\hbar}$, $0<\hbar\ll1$ of stationary Gaussian random smooth functions on the $m$-dimensional torus $\mathbb{T}^{m}:=\mathbb{R}^{m}/\mathbb{Z}^{m}$ approaching the white noise as $\hbar\to0$. We show that there exists universal constants $c_{1},c_{2}>0$ such that for any cube $B\subset\mathbb{R}^{m}$ of size $r\leq1/2$, the number of critical points of $X^{\hbar}$ in the region $B\bmod\mathbb{Z}^{m}\subset\mathbb{T}^{m}$ has mean $\sim c_{1}\operatorname{vol}(B)\hbar^{-m}$, variance $\sim c_{2}\operatorname{vol}(B)\hbar^{-m}$, and satisfies a central limit theorem as $\hbar\searrow0$.

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Bernoulli, Volume 24, Number 2 (2018), 1128-1170.

Received: February 2016
Revised: May 2016
First available in Project Euclid: 21 September 2017

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central limit theorem critical points Gaussian Hilbert spaces Gaussian random functions Kac–Rice formula Wiener chaos


Nicolaescu, Liviu I. Critical points of multidimensional random Fourier series: Central limits. Bernoulli 24 (2018), no. 2, 1128--1170. doi:10.3150/16-BEJ874.

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