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Federico Dalmao

Publications and source records attributed to Federico Dalmao.

At least 19 recordsLinked to original sources
arXiv2026

Multivariable CLT for critical points

We prove a multivariate central limit theorem for the numbers of critical points with all possible indexes which lie above a level of a non-necessarily isotropic Gaussian random field. We prove the non-degeneracy of the limit joint distribution in the isotropic case. We also consider the degenerate case when the value is not restricted to lie above a level. We extend, to the non-isotropic framework, known results by Estrade \& Le{ó}n and Nicolaescu for the Euler characteristic of an excursion set and for the total number of critical points of Gaussian random fields. Though we use the classical tools of chaotic expansions and fourth moment theorem, our proof of the non-degeneracy of the limit distribution does not focus on the explicit description of the lowest order chaotic components but it transforms some convenient Hermite coefficients of arbitrary order into functions of the eigenvalues of a Gaussian Orthogonal Ensemble (GOE) random matrix and use the classical Laplace Method to conclude.

math.PR↗
arXiv2026

Non-asymptotic Tail Bounds for the Kostlan--Shub--Smale Field: Tensor PCA and Spherical $k$-Spin Complexity

This paper builds a hierarchy of explicit, non-asymptotic tail bounds for the supremum of the Kostlan--Shub--Smale (KSS) random field on the sphere, and applies it to two problems: Spiked Tensor PCA and the landscape of the spherical $k$-spin model. For Tensor PCA, we study the non-asymptotic statistical limits of estimating a rank-$R$ symmetric signal tensor of order~$k\ge 3$ and dimension~$d\ge 3$ from a single Gaussian observation at signal-to-noise ratio~$λ$, through the \emph{profile maximum likelihood estimator}, the MLE restricted to normalized rank-$R$ tensors of coherence at least~$κ$. Our analysis uses a single reduction: a deterministic geometric inequality (the Tube Method) and a rank-reduction step bound the estimation error by the supremum of the canonical KSS field, which the Kac--Rice formula turns into a Gaussian integral against the expected absolute characteristic polynomial of a shifted Gaussian Orthogonal Ensemble, controlled in turn by the four explicit tail bounds of our hierarchy (three from a Mehta--Fyodorov representation, one from a Ben Arous--Dembo--Guionnet large deviation). The same reduction yields two results, each with explicit constants. For estimation, a finite-$(k,d)$ error bound recovers the asymptotically optimal rate~$\sqrt{d\log k}$ of Perry, Wein and Bandeira, with explicit dependence on the rank~$R$ and the coherence~$κ$. For the landscape, a two-sided non-asymptotic bracketing of the annealed complexity of the spherical $k$-spin Hamiltonian recovers the Auffinger--Ben Arous--Černý complexity function in the high-dimensional limit.

math.ST↗
arXiv2025

Gaussian random field's anisotropy using excursion sets

This paper addresses the problem of detecting and estimating the anisotropy of a stationary real-valued random field from a single realization of one of its excursion sets. This setting is challenging as it relies on observing a binary image without prior knowledge of the field's mean, variance, or the specific threshold value. Our first contribution is to propose a generalization of Cabaña's contour method to arbitrary dimensions by analyzing the Palm distribution of normal vectors along the excursion set boundaries. We demonstrate that the anisotropy parameters can be recovered by solving a smooth and strongly convex optimization problem involving the eigenvalues of the empirical covariance matrix of these normal vectors. Our second main contribution is a new, model-agnostic statistical test for isotropy in dimension two. We introduce a statistic based on the contour method which is asymptotically distributed as a chi-squared variable with two degrees of freedom under the null hypothesis of quasi-isotropy. Unlike existing methods based on Lipschitz-Killing curvatures, this procedure does not require knowledge of the random field's covariance structure. Extensive numerical experiments show that our test is well-calibrated and more powerful than model-based alternatives as well as that the estimation of the anisotropy parameters, including the directions, is robust and efficient. Finally, we apply this framework to test the quasi-isotropy of the Cosmic Microwave Background (CMB) using the Planck data release 3 mission.

stat.ME↗
arXiv2025

Second Maximum of a Gaussian Random Field and Exact (t-)Spacing test

In this article, we introduce the novel concept of the second maximum of a Gaussian random field on a Riemannian submanifold. This second maximum serves as a powerful tool for characterizing the distribution of the maximum. By utilizing an ad-hoc Kac Rice formula, we derive the explicit form of the maximum's distribution, conditioned on the second maximum and some regressed component of the Riemannian Hessian. This approach results in an exact test, based on the evaluation of spacing between these maxima, which we refer to as the spacing test. We investigate the applicability of this test in detecting sparse alternatives within Gaussian symmetric tensors, continuous sparse deconvolution, and two-layered neural networks with smooth rectifiers. Our theoretical results are supported by numerical experiments, which illustrate the calibration and power of the proposed tests. More generally, this test can be applied to any Gaussian random field on a Riemannian manifold, and we provide a general framework for the application of the spacing test in continuous sparse kernel regression. Furthermore, when the variance-covariance function of the Gaussian random field is known up to a scaling factor, we derive an exact Studentized version of our test, coined the $t$-spacing test. This test is perfectly calibrated under the null hypothesis and has high power for detecting sparse alternatives.

math.ST↗
arXiv2023

A total variation version of Breuer--Major Central Limit Theorem under $\mathbb{D}^{1,2}$ assumption

In this note, we establish a qualitative total variation version of Breuer--Major Central Limit Theorem for a sequence of the type $\frac{1}{\sqrt{n}} \sum_{1\leq k \leq n} f(X_k)$, where $(X_k)_{k\ge 1}$ is a centered stationary Gaussian process, under the hypothesis that the function $f$ has Hermite rank $d \geq 1$ and belongs to the Malliavin space $\mathbb D^{1,2}$. This result in particular extends the recent works of [NNP21], where a quantitative version of this result was obtained under the assumption that the function $f$ has Hermite rank $d= 2$ and belongs to the Malliavin space $\mathbb D^{1,4}$. We thus weaken the $\mathbb D^{1,4}$ integrability assumption to $\mathbb D^{1,2}$ and remove the restriction on the Hermite rank of the base function. While our method is still based on Malliavin calculus, we exploit a particular instance of Malliavin gradient called the sharp operator, which reduces the desired convergence in total variation to the convergence in distribution of a bidimensional Breuer--Major type sequence.

math.PR↗
arXiv2023

On the number of roots of Sturm-Liouville random sums

We consider the number of roots of linear combinations of a system of $n$ orthogonal eigenfunctions of a Sturm-Liouville initial value problem with i.i.d. standard Gaussian coefficients. We prove that its distribution inherits the asymptotic behavior of the number of roots of Quall's random trigonometric polynomials. This result can be thought as a robustness result for the central limit theorem for the number of roots of Quall's random trigonometric polynomials in the sense that small uniform perturbations of sines and cosines do not change the limit distribution.

math.PR↗
arXiv2023

Phase Singularities in Complex Arithmetic Random Waves

Complex arithmetic random waves are stationary Gaussian complex-valued solutions of the Helmholtz equation on the two-dimensional flat torus. We use Wiener-Itô chaotic expansions in order to derive a complete characterization of the second order high-energy behaviour of the total number of phase singularities of these functions. Our main result is that, while such random quantities verify a universal law of large numbers, they also exhibit non-universal and non-central second order fluctuations that are dictated by the arithmetic nature of the underlying spectral measures. Such fluctuations are qualitatively consistent with the cancellation phenomena predicted by Berry (2002) in the case of complex random waves on compact planar domains. Our results extend to the complex setting recent pathbreaking findings by Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013) and Marinucci, Peccati, Rossi and Wigman (2016). The exact asymptotic characterization of the variance is based on a fine analysis of the Kac-Rice kernel around the origin, as well as on a novel use of combinatorial moment formulae for controlling long-range weak correlations.

math.PR↗
arXiv2022

A note on $3d$-monochromatic random waves and cancellation

In this note we prove that the asymptotic variance of the nodal length of complex-valued monochromatic random waves restricted to an increasing domain in $\R^3$ is linear in the volume of the domain. Put together with previous results this shows that a Central Limit Theorem holds true for $3$-dimensional monochromatic random waves. We compare with the variance of the nodal length of the real-valued $2$-dimensional monochromatic random waves where a faster divergence rate is observed, this fact is connected with Berry's cancellation phenomenon. Moreover, we show that a concentration phenomenon takes place.

math.PR↗
arXiv2021

Studying the winding number for stationary Gaussian processes using real variables

We consider the winding number of planar stationary Gaussian processes defined on the line. Under mild conditions, we obtain the asymptotic variance and the Central Limit Theorem for the winding number as the time horizon tends to infinity. In the asymptotic regime, our discrete approach is equivalent to the continuous one studied previously in the literature and our main result extends the existing ones. Our model allows for a general dependence of the coordinates of the process and non-differentiability of one of them. Furthermore, beyond our general framework, we consider as examples an approximation to the winding number of a process whose coordinates are both non-differentiable and the winding number of a process which is not exactly stationary.

math.PR↗
arXiv2021

Central Limit Theorem for the volume of the zero set of Kostlan-Shub-Smale random polynomial systems

We state the Central Limit Theorem, as the degree goes to infinity, for the normalized volume of the zero set of a rectangular Kostlan-Shub-Smale random polynomial system. This paper is a continuation of {\it Central Limit Theorem for the number of real roots of Kostlan Shub Smale random polynomial systems} by the same authors in which the square case was considered. Our main tools are the Kac-Rice formula for the second moment of the volume of the zero set and an expansion of this random variable into the Itô-Wiener Chaos.

math.PR↗
arXiv2020

On the finiteness of the moments of the measure of level sets of random fields

General conditions on smooth real valued random fields are given that ensure the finiteness of the moments of the measure of their level sets. As a by product a new generalized Kac-Rice formula (KRF) for the expectation of the measure of these level sets is obtained when the second moment can be uniformly bounded. The conditions involve (i) the differentiability of the trajectories up to a certain order k, (ii) the finiteness of the moments of the k-th partial derivatives of the field up to another order, (iii) the boundedness of the joint density of the field and some of its derivatives. Particular attention is given to the shot noise processes and fields. Other applications include stationary Gaussian processes, Chi-square processes and regularized diffusion processes. AMS2000 Classifications: Primary 60G60. Secondary 60G15.

math.PR↗
arXiv2019

On 3-dimensional Berry's model

This work aims to study the dislocation or nodal lines of 3D Berry's random wave model. Their expected length is computed both in the isotropic and anisotropic cases, being them compared. Afterwards, in the isotropic case the asymptotic variance and distribution of the length are obtained as the domain grows to the whole space. Under some integrability condition on the covariance function, a central limit theorem is established. The study includes the Berry's monochromatic random waves, the Bargmann-Fock model and the Black-Body radiation as well as a power law model that exhibits an unusual asymptotic behaviour and yields a non-central limit theorem.

math.PR↗
arXiv2018

Central Limit Theorem for the number of real roots of Kostlan Shub Smale random polynomial systems

We obtain a Central Limit Theorem for the normalized number of real roots of a square Kostlan-Shub-Smale random polynomial system of any size as the degree goes to infinity. A study of the asymptotic variance of the number of roots is needed, this result was obtained in [2]. Afterwards we represent the number of roots as an explicit non linear functional belonging to the It{ô}-Wiener chaos. This representation provides a tool for applying the Fourth Moment Theorem and henceforth the asymptotic normality. MSC 2010 subject classifications: Primary 60F05, 30C15, ; secondary 60G60, 65H10.

math.PR↗
arXiv2017

On the real zeros of random trigonometric polynomials with dependent coefficients

We consider random trigonometric polynomials of the form \[ f_n(t):=\sum_{1\le k \le n} a_{k} \cos(kt) + b_{k} \sin(kt), \] whose entries $(a_{k})_{k\ge 1}$ and $(b_{k})_{k\ge 1}$ are given by two independent stationary Gaussian processes with the same correlation function $ρ$. Under mild assumptions on the spectral function $ψ_ρ$ associated with $ρ$, we prove that the expectation of the number $N_n([0,2π])$ of real roots of $f_n$ in the interval $[0,2π]$ satisfies \[ \lim_{n \to +\infty} \frac{\mathbb E\left [N_n([0,2π])\right]}{n} = \frac{2}{\sqrt{3}}. \] The latter result not only covers the well-known situation of independent coefficients but allow us to deal with long range correlations. In particular it englobes the case where the random coefficients are given by a fractional Brownian noise with any Hurst parameter.

math.PR↗
arXiv2015

Local universality of the number of zeros of random trigonometric polynomials with continuous coefficients

Let $X_N$ be a random trigonometric polynomial of degree $N$ with iid coefficients and let $Z_N(I)$ denote the (random) number of its zeros lying in the compact interval $I\subset\mathbb{R}$. Recently, a number of important advances were made in the understanding of the asymptotic behaviour of $Z_N(I)$ as $N\to\infty$, in the case of standard Gaussian coefficients. The main theorem of the present paper is a universality result, that states that the limit of $Z_N(I)$ does not really depend on the exact distribution of the coefficients of $X_N$. More precisely, assuming that these latter are iid with mean zero and unit variance and have a density satisfying certain conditions, we show that $Z_N(I)$ converges in distribution toward $Z(I)$, the number of zeros within $I$ of the centered stationary Gaussian process admitting the cardinal sine for covariance function.

math.PR↗