## Bernoulli

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

### Critical points of multidimensional random Fourier series: Central limits

Liviu I. Nicolaescu

#### 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
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

#### 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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