arXiv subjects

Jürgen Angst

Publications and source records attributed to Jürgen Angst.

At least 19 recordsLinked to original sources

arXiv2026

Global universality of the expected number of zeros of non-analytic random signals

We study the asymptotics as $n$ goes to infinity of $\mathbb E\left[\mathcal{N}(S_n,[0,2π])\right]$, the expected number of zeros in $[0, 2π]$ of a random periodic signal $S_n$ of the form \[ S_n(t)=\sum_{k=1}^{n}a_k f(kt), \] where $f$ is a non-analytic $2π-$periodic function and the coefficients $(a_k)$ are i.i.d. random variables, centered with unit variance. We show in particular that if $a_1$ admits a finite third moment and if the function $f$ is piecewise polynomials and of class $\mathcal C^{7}$, then we have the following universal asymptotics, independent of the particular law of the coefficients $(a_k)$ \[ \lim_{n \to +\infty}\frac{\Esp\left[\mathcal{N}(S_n,[0,2π])\right]}{n}= \frac{2}{\sqrt{3}}\sqrt{\frac{\|f'\|_{L^2([0,2π])}}{\|f\|_{L^2([0,2π])}}}. \] This result thus extends in expectation and at the scale of the whole period $[0,2π]$ the local universality property established {in [Angst-Poly, IMRN, 2019]}, in distribution and in shrinking intervals of size $1/n$. Moreover, it generalizes to a non-analytic context the global universality results obtained in the more classical frameworks of random trigonometric polynomials or random analytic functions. Our approach combines a new almost sure Central Limit Theorem à la Salem--Zygmund for the function $S_n$ when evaluated at a uniform random point in $[0, 2π]$, and as well as suitable uniform integrability and anti-concentration estimates.

math.PR↗

arXiv2026

Convergence of higher derivatives of random polynomials with independent roots

Let $μ$ be a probability measure on $\mathbb C$, and let $P_n$ be the random polynomial whose zeros are sampled independently from $μ$. We study the asymptotic distribution of zeros of high-order derivatives of $P_n$. We show that, for large classes of measures $μ$, the empirical distribution of zeros of the $k$-th derivative converges back to $μ$ for all derivative orders $k=o(n/\log n)$. This includes all discrete measures and a broad family of measures satisfying a mild dimension-nondegeneracy condition. We further establish a robustness result showing that, for arbitrary $μ$, even after adding a vanishing proportion of roots drawn from a dimension-nondegenerate perturbation, the derivative zero measures still converge back to $μ$. These results break the previously known logarithmic barrier on the order of differentiation and demonstrate that the limiting root distribution is preserved under differentiation of order growing nearly linearly with the degree.

math.PR↗

arXiv2024

Roots of random trigonometric polynomials with general dependent coefficients

We consider random trigonometric polynomials with general dependent coefficients. We show that under mild hypotheses on the structure of dependence, the asymptotics as the degree goes to infinity of the expected number of real zeros coincides with the independent case. To the best of our knowledge, this universality result is the first obtained in a non-Gaussian dependent context. Our proof highlights the robustness of real zeros, even in the presence of dependencies. These findings bring the behavior of random polynomials closer to real-world models, where dependencies between coefficients are common.

math.PR↗

arXiv2024

Sharp total variation rates of convergence for fluctuations of linear statistics of $β$-ensembles

In this article, we revisit the question of fluctuations of linear statistics of beta ensembles in the single cut and non-critical regime for general potentials $V$ under mild regularity and growth assumptions. Our main objective is to establish sharp quantitative Central Limit Theorems (CLT) for strong distances, such as the total variation distance, which to the best of our knowledge, is new for general potentials, even qualitatively. Namely, setting $μ_V$ the equilibrium measure, for a test function $ξ\in \mathscr{C}^{14}$, we establish the convergence in total variation of $X_n=\sum_{i=1}^n ξ(λ_i)-n\langle ξ,μ_V\rangle$ to an explicit Gaussian variable at the sharp speed $1/n$. Under the same assumptions, we also establish multivariate CLTs for vectors of linear statistics in $p-$Wasserstein distances for any $p\ge 1$, with the optimal rate $1/n$, a result which already in dimension one sharpens the speed of convergence established in the recent contribution [26] as well as the required regularity on the test functions. A second objective of this paper, in a more qualitative direction, is to establish the so-called super-convergence of linear statistics, that is to say the convergence of all derivatives of the densities of $X_n$ uniformly on $\mathbb{R}$, provided that $ξ\in\mathscr{C}^\infty(\mathbb{R})$ and is not too degenerated in some sense.

math.PR↗

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

Almost sure behavior of the critical points of random polynomials

Let $(Z_k)_{k\geq 1}$ be a sequence of independent and identically distributed complex random variables with common distribution $μ$ and let $P_n(X):=\prod_{k=1}^n (X-Z_k)$ the associated random polynomial in $\mathbb C[X]$. In [Kab15], the author established the conjecture stated by Pemantle and Rivin in [PR13] that the empirical measure $ν_n$ associated with the critical points of $P_n$ converges weakly in probability to the base measure $μ$. In this note, we establish that the convergence in fact holds in the almost sure sense. Our result positively answers a question raised by Z. Kabluchko and formalized as a conjecture in the recent paper [MV22].

math.PR↗

arXiv2021

Fluctuations in Salem--Zygmund almost sure central limit theorem

Let us consider i.i.d. random variables $\{a_k,b_k\}_{k \geq 1}$ defined on a common probability space $(Ω, \mathcal F, \mathbb P)$, following a symmetric Rademacher distribution and the associated random trigonometric polynomials $S_n(θ)= \frac{1}{\sqrt{n}} \sum_{k=1}^n a_k \cos(kθ)+b_k \sin(kθ)$. A seminal result by Salem and Zygmund ensures that $\mathbb{P}-$almost surely, $\forall t\in\mathbb{R}$ \[ \lim_{n \to +\infty} \frac{1}{2π}\int_0^{2π} e^{i t S_n(θ)}dθ=e^{-t^2/2}. \] This result was then further generalized in various directions regarding the coefficients distribution, their dependency structure or else the dimension and the nature of the ambient manifold. To the best of our knowledge, the natural question of the fluctuations in the above limit has not been tackled so far and is precisely the object of this article. Namely, for general i.i.d. symmetric random coefficients having a finite sixth-moment and for a large class of continuous test functions $ϕ$ we prove that \[ \sqrt{n}\left(\frac{1}{2π}\int_0^{2π} ϕ(S_n(θ))dθ-\int_{\mathbb{R}}ϕ(t)\frac{e^{-\frac{t^2}{2}}dt}{\sqrt{2π}}\right)\xrightarrow[n\to\infty]{\text{Law}}~\mathcal{N}\left(0,σ_ϕ^2+\frac{c_2(ϕ)^2}{2}\left(\mathbb{E}(a_1^4)-3\right)\right). \] Here, the constant $σ_ϕ^2$ is explicit and corresponds to the limit variance in the case of Gaussian coefficients and $c_2(ϕ)$ is the coefficient of order $2$ in the decomposition of $ϕ$ in the Hermite polynomial basis. Surprisingly, it thus turns out that the fluctuations are not universal since they both involve the kurtosis of the coefficients and the second coefficient of $ϕ$ in the Hermite basis.

math.PR↗

arXiv2021

Real zeros of random trigonometric polynomials with dependent coefficients

We further investigate the relations between the large degree asymptotics of the number of real zeros of random trigonometric polynomials with dependent coefficients and the underlying correlation function. We consider trigonometric polynomials of the form \[ f_n(t):= \frac{1}{\sqrt{n}}\sum_{k=1}^{n}a_k \cos(kt)+b_k\sin(kt), ~x\in [0,2π], \] where the sequences $(a_k)_{k\geq 1}$ and $(b_k)_{k\geq 1}$ are two independent copies of a stationary Gaussian process centered with variance one and correlation function $ρ$ with associated spectral measure $μ_ρ$. We focus here on the case where $μ_ρ$ is not purely singular and we denote by $ψ_ρ$ its density component with respect to the Lebesgue measure $λ$. Quite surprisingly, we show that the asymptotics of the number of real zeros $\mathcal{N}(f_n,[0,2π])$ of $f_n$ in $[0,2π]$ is not related to the decay of the correlation function $ρ$ but instead to the Lebesgue measure of the vanishing locus of $ψ_ρ$. Namely, assuming that $ψ_ρ$ is $\mathcal{C}^1$ with Hölder derivative on an open set of full measure, one establishes that \[ \lim_{n \to +\infty} \frac{\mathbb E\left[\mathcal{N}(f_n,[0,2π])\right]}{n}= \frac{λ(\{ψ_ρ=0\})}{π\sqrt{2}} + \frac{2π- λ(\{ψ_ρ=0\})}{π\sqrt{3}}. \] On the other hand, assuming a sole log-integrability condition on $ψ_ρ$, which implies that it is positive almost everywhere, we recover the asymptotics of the independent case, i.e. the limit is $\frac{2}{\sqrt{3}}$. Besides, with further assumptions of regularity and existence of negative moment for $ψ_ρ$, we moreover show that the above convergence in expectation can be strengthened to an almost sure convergence.

math.PR↗

arXiv2019

Variations on Salem--Zygmund results for random trigonometric polynomials. Application to almost sure nodal asymptotics

On a probability space $(Ω, \mathcal F, \mathbb P)$ we consider two independent sequences $(a_k)_{k \geq 1}$ and $(b_k)_{k \geq 1}$ of i.i.d. random variables that are centered with unit variance and which admit a moment strictly higher than two. We define the associated random trigonometric polynomial \[ f_n(t) :=\frac{1}{\sqrt{n}} \sum_{k=1}^n a_k \cos(kt)+b_k \sin(kt), \quad t \in \mathbb R. \] In their seminal work, for Rademacher coefficients, Salem and Zygmund showed that $\mathbb P$ almost surely: \[ \forall t\in\mathbb R,\quad \frac{1}{2π}\int_{0}^{2π} \exp\left(i t f_n(x)\right) dx \xrightarrow[n\to\infty]~e^{-\frac{t^2}{2}}. \] In other words, if $X$ denotes an independent random variable uniformly distributed on $[0,2π]$, $\mathbb{P}$ almost surely, under the law of $X$, $f_n(X)$ converges in distribution to a standard Gaussian variable. In this paper, we revisit the above result from different perspectives. Namely, i) we establish a possibly sharp convergence rate for some adequate metric via the Stein's method, ii) we prove a functional counterpart of Salem--Zygmund CLT, iii) we extend it to more general distributions for $X$, iv) we also prove that the convergence actually holds in total variation. As an application, in the case where the random coefficients have a symmetric distribution and admit a moment of order $4$, we show that $\mathbb{P}$ almost surely, for any interval $[a,b] \subset [0, 2π]$ \[\frac{\mathcal N(f_n,[a,b])}{n} \xrightarrow[n \to +\infty]{} \frac{(b-a)}{π\sqrt{3}},\] where $\mathcal N(f_n,[a,b])$ denotes the number of real zeros of $f_n$ in the interval $[a,b]$. To the best of our knowledge, such an almost sure result is new in the framework of random trigonometric polynomials, even in the case of Gaussian coefficients.

math.PR↗

arXiv2019

On the zeros of non-analytic random periodic signals

In this paper, we investigate the local universality of the number of zeros of a random periodic signal of the form $S_n(t)=\sum_{k=1}^n a_k f(k t)$, where $f$ is a $2π-$periodic function satisfying weak regularity conditions and where the coefficients $a_k$ are i.i.d. random variables, that are centered with unit variance. In particular, our results hold for continuous piecewise linear functions. We prove that the number of zeros of $S_n(t)$ in a shrinking interval of size $1/n$ converges in law as $n$ goes to infinity to the number of zeros of a Gaussian process whose explicit covariance only depends on the function $f$ and not on the common law of the random coefficients $(a_k)$. As a byproduct, this entails that the point measure of the zeros of $S_n(t)$ converges in law to an explicit limit on the space of locally finite point measures on $\mathbb R$ endowed with the vague topology. The standard tools involving the regularity or even the analyticity of $f$ to establish such kind of universality results are here replaced by some high-dimensional Berry-Esseen bounds recently obtained in [CCK17]. The latter allow us to prove functional CLT's in $C^1$ topology in situations where usual criteria can not be applied due to the lack of regularity.

math.PR↗

arXiv2018

On the Poisson boundary of the relativistic Brownian motion

In this paper, we determine the Poisson boundary of the relativistic Brownian motion in two classes of Lorentzian manifolds, namely model manifolds of constant scalar curvature and Robertson--Walker space-times, the latter constituting a large family of curved manifolds. Our objective is two fold: on the one hand, to understand the interplay between the geometry at infinity of these manifolds and the asymptotics of random sample paths, in particular to compare the stochastic compactification given by the Poisson boundary to classical purely geometric compactifications such as the conformal or causal boundaries. On the other hand, we want to illustrate the power of the dévissage method introduced by the authors in [AT16], method which we show to be particularly well suited in the geometric contexts under consideration here.

math.PR↗

arXiv2018

On the absolute continuity of random nodal volumes

We study the absolute continuity with respect to the Lebesgue measure of the distribution of the nodal volume associated with a smooth, non-degenerated and stationary Gaussian field $(f(x), {x \in \mathbb R^d})$. Under mild conditions, we prove that in dimension $d\geq 3$, the distribution of the nodal volume has an absolutely continuous component plus a possible singular part. This singular part is actually unavoidable baring in mind that some Gaussian processes have a positive probability to keep a constant sign on some compact domain. Our strategy mainly consists in proving closed Kac--Rice type formulas allowing one to express the volume of the set $\{f =0\}$ as integrals of explicit functionals of $(f,\nabla f,\text{Hess}(f))$ and next to deduce that the random nodal volume belongs to the domain of a suitable Malliavin gradient. The celebrated Bouleau--Hirsch criterion then gives conditions ensuring the absolute continuity.

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↗

arXiv2016

Universality of the nodal length of bivariate random trigonometric polynomials

We consider random trigonometric polynomials of the form \[ f_n(x,y)=\sum_{1\le k,l \le n} a_{k,l} \cos(kx) \cos(ly), \] where the entries $(a_{k,l})_{k,l\ge 1}$ are i.i.d. random variables that are centered with unit variance. We investigate the length $\ell_K(f_n)$ of the nodal set $Z_K(f_n)$ of the zeros of $f_n$ that belong to a compact set $K \subset \mathbb R^2$. We first establish a local universality result, namely we prove that, as $n$ goes to infinity, the sequence of random variables $n\, \ell_{K/n}(f_n)$ converges in distribution to a universal limit which does not depend on the particular law of the entries. We then show that at a macroscopic scale, the expectation of $\ell_{[0,π]^2}(f_n)/n$ also converges to an universal limit. Our approach provides two main byproducts: (i) a general result regarding the continuity of the volume of the nodal sets with respect to $C^1$-convergence which refines previous findings of Rusakov et al., Iksanov et al. and Azaïs et al., and (ii) a new strategy for proving small ball estimates in random trigonometric models, providing in turn uniform local controls of the nodal volumes.

math.PR↗

arXiv2015

Universality of the mean number of real zeros of random trigonometric polynomials under a weak Cramer condition

We investigate the mean number of real zeros over an interval $[a,b]$ of a random trigonometric polynomial of the form $\sum_{k=1}^n a_k \cos(kt)+b_k \sin(kt)$ where the coefficients are i.i.d. random variables. Under mild assumptions on the law of the entries, we prove that this mean number is asymptotically equivalent to $\frac{n(b-a)}{π\sqrt{3}}$ as $n$ goes to infinity, as in the known case of standard Gaussian coefficients. Our principal requirement is a new Cramer type condition on the characteristic function of the entries which does not only hold for all continuous distributions but also for discrete ones in a generic sense. To our knowledge, this constitutes the first universality result concerning the mean number of zeros of random trigonometric polynomials. Besides, this is also the first time that one makes use of the celebrated Kac-Rice formula not only for continuous random variables as it was the case so far, but also for discrete ones. Beyond the proof of a non asymptotic version of Kac-Rice formula, our strategy consists in using suitable small ball estimates and Edgeworth expansions for the Kolmogorov metric under our new weak Cramer condition, which both constitute important byproducts of our approach.

math.PR↗

arXiv2015

Kinetic Brownian motion on Riemannian manifolds

We consider in this work a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle $T^1 \mathcal M$ of a Riemannian manifold $(\mathcal M,g)$, collectively called kinetic Brownian motions, that are random perturbations of the geodesic flow, with a parameter $σ$ quantifying the size of the noise. Projection on $\mathcal M$ of these processes provides random $C^1$ paths in $\mathcal M$. We show, both qualitively and quantitatively, that the laws of these $\mathcal M$-valued paths provide an interpolation between geodesic and Brownian motions. This qualitative description of kinetic Brownian motion as the parameter $σ$ varies is complemented by a thourough study of its long time asymptotic behaviour on rotationally invariant manifolds, when $σ$ is fixed, as we are able to give a complete description of its Poisson boundary in geometric terms.

math.PR↗

arXiv2014

Asymptotic behavior of a relativistic diffusion in Robertson-Walker space-times

We determine the long-time asymptotic behavior of a relativistic diffusion taking values in the unitary tangent bundle of a Robertson-Walker space-time. We prove in particular that when approaching the explosion time of the diffusion, its projection on the base manifold almost surely converges to a random point of the causal boundary and we also describe the behavior of the tangent vector in the neighborhood of this limiting point.

math.PR↗

arXiv2014

Poisson boundary of a relativistic diffusion in curved space-times: an example

We study in details the long-time asymptotic behavior of a relativistic diffusion taking values in the unitary tangent bundle of a curved Lorentzian manifold, namely a spatially flat and fast expanding Robertson-Walker space-time. We prove in particular that the Poisson boundary of the diffusion can be identified with the causal boundary of the underlying manifold.

math.PR↗