Bernoulli

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

Critical points of multidimensional random Fourier series: Central limits

Liviu I. Nicolaescu

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Abstract

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 $\hbar\searrow0$.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.bj/1505980892

Digital Object Identifier
doi:10.3150/16-BEJ874

Mathematical Reviews number (MathSciNet)
MR3706790

Zentralblatt MATH identifier
06778361

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

Citation

Nicolaescu, Liviu I. Critical points of multidimensional random Fourier series: Central limits. Bernoulli 24 (2018), no. 2, 1128--1170. doi:10.3150/16-BEJ874. https://projecteuclid.org/euclid.bj/1505980892


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