We compute the variance asymptotics for the number of real zeros of trigonometric polynomials with random dependent Gaussian coefficients and show that under mild conditions, the asymptotic behavior is the same as in the independent framework. In fact our proof goes beyond this framework and makes explicit the variance asymptotics of various models of random Gaussian polynomials. Though we use the Kac--Rice formula, we do not use the explicit closed formula for the second moment of the number of zeros, but we rather rely on intrinsic properties of the Kac--Rice density.
In this paper, we study the number of real roots of random trigonometric polynomials with iid coefficients. When the coefficients have zero mean, unit variance and some finite high moments, we show that the variance of the number of real roots is asymptotically linear in terms of the expectation; furthermore, the multiplicative constant in this linear relationship depends only on the kurtosis of the common distribution of the polynomial's coefficients. This result is in sharp contrast to the classical Kac polynomials whose corresponding variance depends only on the first two moments. Our resul…
We prove that the minimum of the modulus of a random trigonometric polynomial with Gaussian coefficients, properly normalized, has limiting exponential distribution.
We deduce bilateral interrelations between Fourier coefficients for lacunar trigonometric series and modulus of their continuity. We obtain also as an application some conditions for continuity and discontinuity for Gaussian periodic stationary random centered processes.
In this article, we consider the following family of random trigonometric polynomials $p_n(t,Y)=\sum_{k=1}^n Y_{k,1} \cos(kt)+Y_{k,2}\sin(kt)$ for a given sequence of i.i.d. random variables $\{Y_{k,1},Y_{k,2}\}_{k\ge 1}$ which are centered and standardized. We set $\mathcal{N}([0,π],Y)$ the number of real roots over $[0,π]$ and $\mathcal{N}([0,π],G)$ the corresponding quantity when the coefficients follow a standard Gaussian distribution. We prove under a Doeblin's condition on the distribution of the coefficients that $$ \lim_{n\to\infty}\frac{\text{Var}\left(\mathcal{N}_n([0,π],Y)\right)}{n…
We prove a Central Limit Theorem for the number of zeros of random trigonometric polynomials of the form $K^{-1/2}\sum_{n=1}^{K} a_n\cos(nt)$, being $(a_n)_n$ independent standard Gaussian random variables. In particular, we prove the conjecture by Farahmand, Granville & Wigman that the variance is equivalent to $V^2K$, $0<V^2<\infty$, as $K\to\infty$. % The case of stationary trigonometric polynomials was studied by Granville & Wigman and by Aza\"\is & León. Our approach is based on the Hermite/Wiener-Chaos decomposition for square-integrable functionals of a Gaussian process and on Rice Form…
The large degree asymptotics of the expected number of real zeros of a random trigonometric polynomial $$ T_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) + b_j \sin (j x), \ x \in (0,2π), $$ with i.i.d. real-valued standard Gaussian coefficients is known to be $ 2n / \sqrt{3} $. In this article, we consider quite a different and extreme setting on the set of the coefficients of $ T_n $. We show that a random trigonometric polynomial of degree $ n $ with $ \ell $-periodic i.i.d. Gaussian coefficients is expected to have significantly more real zeros compared to the classical case with i.i.d. Gaussian …
We consider random trigonometric polynomials of the form \[ f_n(t):=\frac{1}{\sqrt{n}} \sum_{k=1}^{n}a_k \cos(k t)+b_k \sin(k t), \] where $(a_k)_{k\geq 1}$ and $(b_k)_{k\geq 1}$ are two independent stationary Gaussian processes with the same correlation function $ρ: k \mapsto \cos(kα)$, with $α\geq 0$. We show that the asymptotics of the expected number of real zeros differ from the universal one $\frac{2}{\sqrt{3}}$, holding in the case of independent or weakly dependent coefficients. More precisely, for all $\varepsilon>0$, for all $\ell \in (\sqrt{2},2]$, there exists $α\geq 0$ and $n\geq …
We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials $$ X_n(t)=u+\frac{1}{\sqrt{n}}\sum_{k=1}^n (A_k\cos(kt)+B_k\sin(kt)), \quad t\in [0,2π],\quad u\in\mathbb{R} $$ whose coefficients $A_k, B_k$, $k\in\mathbb{N}$, are independent identically distributed random variables with zero mean and unit variance. If $N_n[a, b]$ denotes the number of real roots of $X_n$ in an interval $[a,b]\subseteq [0,2π]$, we prove that $$ \lim_{n\rightarrow\infty} \frac{\mathbb{E} N_n[a,b]}{n}=\frac{b-a}{π\sqrt{3}} e^{-\frac{u^2}{2}}. $$
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…
It has been shown in a recent work by Yakir-Zeitouni that the minimum modulus of random trigonometric polynomials with Gaussian coefficients has a limiting exponential distribution. We show this is a universal phenomenon. Our approach relates the joint distribution of small values of the polynomial at a fixed number $m$ of points on the circle to the distribution of a certain random walk in a $4m$-dimensional phase space. Under Diophantine approximation conditions on the angles, we obtain strong small ball estimates and a local central limit theorem for the distribution of the walk.
In this paper, we quantify the rate of convergence between the distribution of number of zeros of random trigonometric polynomials (RTP) with i.i.d. centered random coefficients and the number of zeros of a stationary centered Gaussian process G, whose covariance function is given by the sinc function. First, we find the convergence of the RTP towards G in the Wasserstein-1 distance, which in turn is a consequence of Donsker Theorem. Then, we use this result to derive the rate of convergence between their respective number of zeros. Since the number of real zeros of the RTP is not a continuous…
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 pa…
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 th…
We prove a uniform generalized gaussian bound for the powers of a discrete convolution operator in one space dimension. Our bound is derived under the assumption that the Fourier transform of the coefficients of the convolution operator is a trigonometric rational function, which generalizes previous results that were restricted to trigonometric polynomials. We also allow the modulus of the Fourier transform to attain its maximum at finitely many points over a period.
With motivation from K. Dȩbicki and P. Kisowski (2007), in this paper we derive the exact tail asymptotics of $α(t)$-locally stationary Gaussian processes with non-constant variance functions. We show that some certain variance functions lead to qualitatively new results.
Consider a random trigonometric polynomial $X_n: \mathbb R \to \mathbb R$ of the form $$ X_n(t) = \sum_{k=1}^n \left( ξ_k \sin (kt) + η_k \cos (kt)\right), $$ where $(ξ_1,η_1),(ξ_2,η_2),\ldots$ are independent identically distributed bivariate real random vectors with zero mean and unit covariance matrix. Let $(s_n)_{n\in\mathbb N}$ be any sequence of real numbers. We prove that as $n\to\infty$, the number of real zeros of $X_n$ in the interval $[s_n+a/n, s_n+ b/n]$ converges in distribution to the number of zeros in the interval $[a,b]$ of a stationary, zero-mean Gaussian process with correla…
Gaussian random processes which variances reach theirs maximum values at unique points are considered. Exact asymptotic behaviors of probabilities of large absolute maximums of theirs trajectories have been evaluated using Double Sum Method under the widest possible conditions.
We present a self-contained proof of a uniform bound on multi-point correlations of trigonometric functions of a class of Gaussian random fields. It corresponds to a special case of the general situation considered in [Hairer-Xu], but with improved estimates. As a consequence, we establish convergence of a class of Gaussian fields composite with more general functions. These bounds and convergences are useful ingredients to establish weak universalities of several singular stochastic PDEs.
In this note, we obtain a Gaussian concentration inequality for a class of non-Lipschitz functions. In the one-dimensional case, our results supplement those established by Paouris and Valettas in [8].
We give a precise measure of the rate at which repeated differentiation of a random trigonometric polynomial causes the roots of the function to approach equal spacing. This can be viewed as a toy model of crystallization in one dimension. In particular we determine the asymptotics of the distribution of the roots around the crystalline configuration and find that the distribution is not Gaussian.
Gaussian random fields on Euclidean spaces whose variances reach their maximum values at unique points are considered. Exact asymptotic behaviors of probabilities of large absolute maximum of theirs trajectories have been evaluated using Double Sum Method under the widest possible conditions.
We investigate the zero set of a stationary Gaussian process on the real line, and in particular give lower bounds for the variance of the number of points on a large interval, in all generality. We prove that this point process is never hyperuniform, i.e. the variance is at least linear, and give necessary conditions to have linear variance, which are close to be sharp. We study the class of symmetric Bernoulli convolutions and give an example where the zero set is super rigid, weakly mixing, and not hyperuniform.
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 \x…
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 al…
We study the asymptotic distribution of the number $Z_{N}$ of zeros of random trigonometric polynomials of degree $N$ as $N\to\infty$. It is known that as $N$ grows to infinity, the expected number of the zeros is asymptotic to $\frac{2}{\sqrt{3}}\cdot N$. The asymptotic form of the variance was predicted by Bogomolny, Bohigas and Leboeuf to be $cN$ for some $c>0$. We prove that $\frac{Z_{N}-\E Z_{N}}{\sqrt{cN}}$ converges to the standard Gaussian. In addition, we find that the analogous result is applicable for the number of zeros in short intervals.
Brown-Resnick processes are max-stable processes that are associated to Gaussian processes. Their simulation is often based on the corresponding spectral representation which is not unique. We study to what extent simulation accuracy and efficiency can be improved by minimizing the maximal variance of the underlying Gaussian process. Such a minimization is a difficult mathematical problem that also depends on the geometry of the simulation domain. We extend Matheron's (1974) seminal contribution in two aspects: (i) making his description of a minimal maximal variance explicit for convex variog…
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…
Let $X(s,t), (s,t)\in E$, with $E\subset \mathbb{R}^2$ a compact set, be a centered two dimensional Gaussian random field with continuous trajectories and variance function $σ(s,t)$. Denote by $\mathcal{L}=\{(s,t): σ(s,t)=\max_{(s',t')\in E}σ(s',t')\}$. In this contribution, we derive the exact asymptotics of $\mathbb{P}\left(\sup_{(s,t)\in E}X(s,t)>u\right)$, as $u\to\infty$, under condition that $\mathcal{L}$ is a smooth curve. We illustrate our findings by an application concerned with extremes of the aggregation of two independent fractional Brownian motions.
Recently, many results have been established drawing a parallel between Bernoulli percolation and models given by levels of smooth Gaussian fields with unbounded, strongly decaying correlation. In a previous work with D. Gayet , we started to extend these analogies by adapting the first basic results of classical first passage percolation in this new framework: positivity of the time constant and the ball-shape theorem. In the present paper, we present a proof inspired by Kesten of other basic properties of the new FPP model: an upper bound on the variance in the FPP pseudometric given by the …
Let $K_n$ be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on $\R^d$. We establish variance asymptotics as $n \to \infty$ for the re-scaled intrinsic volumes and $k$-face functionals of $K_n$, $k \in \{0,1,...,d-1\}$, resolving an open problem. Variance asymptotics are given in terms of functionals of germ-grain models having parabolic grains with apices at a Poisson point process on $\R^{d-1} \times \R$ with intensity $e^h dh dv$. The scaling limit of the boundary of $K_n$ as $n \to \infty$ converges to a festoon of parabolic surfaces, co…
In this paper, we study a random field constructed from the two-dimensional Gaussian free field (GFF) by modifying the variance along the scales in the neighborhood of each point. The construction can be seen as a local martingale transform and is akin to the time-inhomogeneous branching random walk. In the case where the variance takes finitely many values, we compute the first order of the maximum and the log-number of high points. These quantities were obtained by Bolthausen, Deuschel and Giacomin (2001) and Daviaud (2006) when the variance is constant on all scales. The proof relies on a t…
We investigate the asymptotic variance of Gaussian nodal excursions in the Euclidean space, focusing on the case where the spectral measure has incommensurable atoms. This study requires to establish fine recurrence properties in 0 for the associated irrational random walk on the torus. We show in particular that the recurrence magnitude depends strongly on the diophantine properties of the atoms, and the same goes for the variance asymptotics of nodal excursions. More specifically, if the spectral measures contains atoms which ratios are well approximable by rationals, the variance is likely …
The q-Gaussians are discussed from the point of view of variance mixtures of normals and exchangeability. For each q< 3, there is a q-Gaussian distribution that maximizes the Tsallis entropy under suitable constraints. This paper shows that q-Gaussian random variables can be represented as variance mixtures of normals. These variance mixtures of normals are the attractors in central limit theorems for sequences of exchangeable random variables; thereby, providing a possible model that has been extensively studied in probability theory. The formulation provided has the additional advantage of y…
In this article we consider Wigner matrices $X_N$ with variance profiles (also called Wigner-type matrices) which are of the form $X_N(i,j) = σ(i/N,j/N) a_{i,j} / \sqrt{N}$ where $σ$ is a symmetric real positive function of $[0,1]^2$ and $σ$ will be taken either continuous or piecewise constant. We prove a large deviation principle for the largest eigenvalue of those matrices under the same condition of sharp sub-Gaussian bound and for some other assumptions on $σ$. These sub-Gaussian bounds are verified for example for Gaussian variables, Rademacher variables or uniform variables on $[- \sqrt…
We continue our study of the scale-inhomogeneous Gaussian free field introduced in Arguin and Ouimet (2016). Firstly, we compute the limiting free energy on V_N and adapt a technique of Bovier and Kurkova (2004b) to determine the limiting two-overlap distribution. The adaptation was already successfully applied in the simpler case of Arguin and Zindy (2015), where the limiting free energy was computed for the field with two levels (in the center of V_N) and the limiting two-overlap distribution was determined in the homogeneous case. Our results agree with the analogous quantities for the Gene…
Following Wiener, we consider the zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We show that the variance of the number of zeroes in a long horizontal rectangle $[0,T]\times [a,b]$ is asymptotically between $cT$ and $CT^2$, with positive constants $c$ and $C$. We also supply with conditions (in terms of the spectral measure) under which the variance asymptotically grows linearly, as a quadratic function of $T$, or has intermediate growth.
It is well known that the expected number of real zeros of a random cosine polynomial $ V_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) , \ x \in (0,2π) $, with the $ a_j $ being standard Gaussian i.i.d. random variables is asymptotically $ 2n / \sqrt{3} $. On the other hand, some of the previous works on the random cosine polynomials with dependent coefficients show that such polynomials have at least $ 2n / \sqrt{3} $ expected real zeros lying in one period. In this paper we investigate two classes of random cosine polynomials with pairwise equal blocks of coefficients. First, we prove that a rando…
The assumption that the elements of the cost matrix in the classical assignment problem are drawn independently from a standard Gaussian distribution motivates the study of a particular Gaussian field indexed by the symmetric permutation group. The correlation structure of the field is determined by the Hamming distance between two permutations. The expectation of the maximum of the field is shown to go to infinity in the same way as if all variables of the field were independent. However, the variance of the maximum is shown to converge to zero at a rate which is slower than under independenc…
In this paper we consider the problem of a measure that allows us to describe the spatial and temporal dependence structure of multivariate time series with innovations having infinite variance. By using recent results obtained in the problem of temporal dependence structure of univariate stochastic processes, where the auto-codifference was used, we extend its idea and propose a cross-codifference measure for a general vector autoregressive model of order 1 (VAR(1)). Next, we derive an analytical results for VAR(1) model with Gaussian and sub-Gaussian innovations, that are characterized by fi…
We study the variance of the number of zeroes of a stationary Gaussian process on a long interval. We give a simple asymptotic description under mild mixing conditions. This allows us to characterise minimal and maximal growth. We show that a small (symmetrised) atom in the spectral measure at a special frequency does not affect the asymptotic growth of the variance, while an atom at any other frequency results in maximal growth. Our results allow us to analyse a large number of interesting examples.
We investigate the sub-Gaussian property for almost surely bounded random variables. If sub-Gaussianity per se is de facto ensured by the bounded support of said random variables, then exciting research avenues remain open. Among these questions is how to characterize the optimal sub-Gaussian proxy variance? Another question is how to characterize strict sub-Gaussianity, defined by a proxy variance equal to the (standard) variance? We address the questions in proposing conditions based on the study of functions variations. A particular focus is given to the relationship between strict sub-Gaus…
The tube method or the volume-of-tube method approximates the tail probability of the maximum of a smooth Gaussian random field with zero mean and unit variance. This method evaluates the volume of a spherical tube about the index set, and then transforms it to the tail probability. In this study, we generalize the tube method to a case in which the variance is not constant. We provide the volume formula for a spherical tube with a non-constant radius in terms of curvature tensors, and the tail probability formula of the maximum of a Gaussian random field with inhomogeneous variance, as well a…
We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued polynomials (algebraic or trigonometric) of large degree on the sphere or torus, and translation-invariant smooth Gaussian functions on the Euclidean space restricted to large domains.
In this article we study weighted sums of $n$ i.i.d. Gamma($α$) random variables with nonnegative weights. We show that for $n \geq 1/α$ the sum with equal coefficients maximizes differential entropy when variance is fixed. As a consequence, we prove that among nonnegative quadratic forms in $n$ independent standard Gaussian random variables, a diagonal form with equal coefficients maximizes differential entropy, under a fixed variance. This provides a sharp lower bound for the relative entropy between a nonnegative quadratic form and a Gaussian random variable. Bounds on capacities of transmi…
Let $n$ be a large integer, and let $G$ be the standard Gaussian vector in $R^n$. Paouris, Valettas and Zinn (2015) showed that for all $p\in[1,c\log n]$, the variance of the $\ell_p^n$--norm of $G$ is equivalent, up to a constant multiple, to $\frac{2^p}{p}n^{2/p-1}$, and for $p\in[C\log n,\infty]$, $\mathbb{Var}\|G\|_p\simeq (\log n)^{-1}$. Here, $C,c>0$ are universal constants. That result left open the question of estimating the variance for $p$ logarithmic in $n$. In this note, we resolve the question by providing a complete characterization of $\mathbb{Var}\|G\|_p$ for all $p$. We show t…
The intrinsic volumes of Gaussian polytopes are considered. A lower variance bound for these quantities is proved, showing that, under suitable normalization, the variances converge to strictly positive limits. The implications of this missing piece of the jigsaw in the theory of Gaussian polytopes are discussed.
We are concerned with zeros of random power series with coefficients being a stationary, centered, complex Gaussian process. We show that the expected number of zeros in every smooth domain in the disk of convergence is less than that of the hyperbolic GAF with i.i.d. coefficients. When coefficients are finitely dependent, i.e., the spectral density is a trigonometric polynomial, we derive precise asymptotics of the expected number of zeros inside the disk of radius $r$ centered at the origin as $r$ tends to the radius of convergence, in the proof of which we clarify that the negative contribu…
