Random Trigonometric Polynomials Research Articles

Rates of convergence for the number of zeros of random trigonometric polynomials

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 function, we use the Kac-Rice formula to express it as the limit of an integral and, in this way, we approximate it by locally Lipschitz continuous functions.

Real zeros of random trigonometric polynomials with $\ell$-periodic coefficients

Real zeros of random trigonometric polynomials with $\ell$-periodic coefficients

Fluctuations in Salem–Zygmund almost sure Central Limit Theorem

We consider the following family of random trigonometric polynomials of the form Sn(θ):=1 n ∑k=1nakcos(kθ)+bksin(kθ), where the variables {ak,bk}k≥1 are i.i.d. and symmetric, defined on a common probability space (Ω,F,P). In the seminal paper [SZ54], Salem and Zygmund proved that for Rademacher coefficients, P−almost surely and for all t∈R limn→+∞ 1 2π∫02πeitSn(θ)dθ=e−t2∕2. To the best of our knowledge, the natural question of the fluctuations in the above almost sure 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 n1 2π∫02πϕ(S n(θ))dθ−∫Rϕ(t)e−t2 2dt 2π→n→∞LawN0,Σϕ2, where the limit variance is given by Σϕ2:=σϕ2+c2(ϕ)22E[a14]−3. Here, the constant σϕ2 is explicit and corresponds to the limit variance in the case of Gaussian coefficients and c2(ϕ) is the coefficient of order 2 in the decomposition of ϕ in the Hermite polynomial basis. It thus turns out that the fluctuations are Gaussian and not universal since they involve the kurtosis of the random coefficients. The method we develop here is robust and allows to use Wiener chaos expansions for proving CLTs for functionals of general, non-necessarily Gaussian, random fields. It combines contributions from the Malliavin–Stein method, noise stability results, and algebraic considerations around Newton sums and the Newton–Girard formula.

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 \[ f n ( t ) ≔ 1 n ∑ k = 1 n a k cos ⁡ ( k t ) + b k sin ⁡ ( k t ) ,   x ∈ [ 0 , 2 π ] , f_n(t)≔\frac {1}{\sqrt {n}}\sum _{k=1}^{n}a_k \cos (kt)+b_k\sin (kt), ~x\in [0,2\pi ], \] where the sequences ( a k ) k ≥ 1 (a_k)_{k\geq 1} and ( b k ) k ≥ 1 (b_k)_{k\geq 1} are two independent copies of a stationary Gaussian process centered with variance one and correlation function ρ \rho with associated spectral measure μ ρ \mu _{\rho } . We focus here on the case where μ ρ \mu _{\rho } is not purely singular and we denote by ψ ρ \psi _{\rho } its density component with respect to the Lebesgue measure λ \lambda . Quite surprisingly, we show that the asymptotics of the number of real zeros N ( f n , [ 0 , 2 π ] ) \mathcal {N}(f_n,[0,2\pi ]) of f n f_n in [ 0 , 2 π ] [0,2\pi ] is not related to the decay of the correlation function ρ \rho but instead to the Lebesgue measure of the vanishing locus of ψ ρ \psi _{\rho } . Namely, assuming that ψ ρ \psi _{\rho } is C 1 \mathcal {C}^1 with Hölder derivative on an open set of full measure, one establishes that \[ lim n → + ∞ E [ N ( f n , [ 0 , 2 π ] ) ] n = λ ( { ψ ρ = 0 } ) π 2 + 2 π − λ ( { ψ ρ = 0 } ) π 3 . \lim _{n \to +\infty } \frac {\mathbb {E}\left [\mathcal {N}(f_n,[0,2\pi ])\right ]}{n}= \frac {\lambda (\{\psi _{\rho }=0\})}{\pi \sqrt {2}} + \frac {2\pi - \lambda (\{\psi _{\rho }=0\})}{\pi \sqrt {3}}. \] On the other hand, assuming a sole log-integrability condition on ψ ρ \psi _{\rho } , which implies that it is positive almost everywhere, we recover the asymptotics of the independent case: \[ lim n → + ∞ E [ N ( f n , [ 0 , 2 π ] ) ] n = 2 3 . \lim _{n \to +\infty } \frac {\mathbb {E}\left [\mathcal {N}(f_n,[0,2\pi ])\right ]}{n}= \frac {2}{\sqrt {3}}. \] The latter asymptotics thus broadly generalizes the main result of Angst, Dalmao, and Poly [Proc. Amer. Math. Soc. 147 (2019), pp. 205–214] where the spectral density was assumed to be continuous and bounded from below. Besides, with further assumptions of regularity and existence of negative moment for ψ ρ \psi _{\rho } , which encompass e.g. the case of random coefficients being increments of fractional Brownian motion with any Hurst parameter, we moreover show that the above convergence in expectation can be strengthened to an almost sure convergence.

Random trigonometric polynomials: Universality and non-universality of the variance for the number of real roots

Dans cet article, nous étudions le nombre de zéros réels de polynômes trigonométriques avec coefficients i.i.d. Quand les coefficients sont centrés, réduits, et possèdent des moments finis d’ordre suffisamment élevé, nous montrons que la variance du nombre de zéros est asymptotiquement linéaire en son espérance ; de plus, la constante multiplicative dans cette relation linéaire dépend seulement du kurtosis de la loi commune des coefficients du polynôme. Ce résultat contraste fortement avec les classiques polynômes de Kac pour lesquels la variance ne dépend que des deux premiers moments. Il s’agit probablement du premier résultat sur ce type de questions pour des lois générales des coefficients, y compris des lois discrètes, pour des polynômes qui ne sont pas dans la famille des polynômes de Kac. L’expansion de Edgeworth, la formule asymptotique de Kac–Rice et l’analyse précise des fonctions caractéristiques sont les outils principaux de notre approche.

Roots of random functions: A framework for local universality

:We investigate the local distribution of roots of random functions of the form $F_n(z)=\sum_{i=1}^n\xi_i\phi_i(z)$, where $\xi_i$ are independent random variables and $\phi_i(z)$ are arbitrary analytic functions. Starting with the fundamental works of Kac and Littlewood-Offord in the 1940s, random functions of this type have been studied extensively in many fields of mathematics.We develop a robust framework to solve the problem by reducing, via universality theorems, the calculation of the distribution of the roots and the interaction between them to the case where $\xi_i$ are Gaussian. In this special case, one can use the Kac-Rice formula and various other tools to obtain precise answers.Our framework has a wide range of applications, which include the most popular models of random functions, such as random trigonometric polynomials and all basic classes of random algebraic polynomials (Kac, Weyl, and elliptic). Each of these ensembles has been studied heavily by deep and diverse methods. Our method, for the first time, provides a unified treatment for all of them.Among the applications, we derive the first local universality result for random trigonometric polynomials with arbitrary coefficients. When restricted to the study of real roots, this result extends several recent results, proved for less general ensembles. For random algebraic polynomials, we strengthen several recent results of Tao and the second author, with significantly simpler proofs. As a corollary, we sharpen a classical result of Erd\"{o}s and Offord on real roots of Kac polynomials, providing an optimal error estimate. Another application is a refinement of a recent result of Flasche and Kabluchko on the roots of random Taylor series.

The minimum modulus of Gaussian trigonometric polynomials

We prove that the minimum of the modulus of a random trigonometric polynomial with Gaussian coefficients, properly normalized, has limiting exponential distribution.

Variations on Salem–Zygmund results for random trigonometric polynomials: application to almost sure nodal asymptotics

On a probability space $(\Omega, \mathcal F, \mathbb P)$ 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: \[ \forall t\in\mathbb R,\quad \frac{1}{2\pi}\int_{0}^{2\pi} \exp\left(i t f_n(x)\right) dx \xrightarrow[n\to\infty]~e^{-\frac{t^2}{2}}. \] In other words, if $X$ denotes an independent random variable uniformly distributed on $[0,2\pi]$, $\mathbb{P}$ 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 $[a,b] \subset [0, 2\pi]$ \[\frac{\mathcal N(f_n,[a,b])}{n} \xrightarrow[n \to +\infty]{} \frac{(b-a)}{\pi \sqrt{3}},\] 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.

Distribution of Complex Algebraic Numbers on the Unit Circle

For $-\pi\leq\beta_1<\beta_2\leq\pi$ denote by $\Phi_{\beta_1,\beta_2}(Q)$ the number of algebraic numbers on the unit circle with arguments in $[\beta_1,\beta_2]$ of degree $2m$ and with elliptic height at most $Q$. We show that \[ \Phi_{\beta_1,\beta_2}(Q)=Q^{m+1}\int\limits_{\beta_1}^{\beta_2}{p(t)}\,{\rm d}t+O\left(Q^m\,\log Q\right),\quad Q\to\infty, \] where $p(t)$ coincides up to a constant factor with the density of the roots of some random trigonometric polynomial. This density is calculated explicitly using the Edelman--Kostlan formula.

Real zeros of random trigonometric polynomials with pairwise equal blocks of coefficients

It is well known that the expected number of real zeros of a random cosine polynomial Vn(x)= ∑j=0naj cos(jx), x∈(0,2π), with the aj being standard Gaussian i.i.d. random variables, is asymptotically 2n∕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∕3 expected real zeros lying in one period. We investigate two classes of random cosine polynomials with pairwise equal blocks of coefficients. First, we prove that a random cosine polynomial with the blocks of coefficients being of a fixed length and satisfying A2j+1=A2j possesses the same expected real zeros as the classical case. Afterwards, we study a case containing only two equal blocks of coefficients, and show that in this case significantly more real zeros should be expected compared with those of the classical case.

New asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficients

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 $\rho: k \mapsto \cos(k\alpha)$, with $\alpha\geq 0$. 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 $\alpha \geq 0$ and $n\geq 1$ large enough such that $$\left|\frac{\mathbb{E}\left[\mathcal{N}(f_n,[0,2\pi])\right]}{n}-\ell\right|\leq \varepsilon,$$ where $\mathcal N(f_n,[0,2\pi])$ denotes the number of real zeros of the function $f_n$ in the interval $[0,2\pi]$. Therefore, this result provides the first example where the expected number of real zeros do not converge as $n$ goes to infinity by exhibiting a whole range of possible limits ranging from $\sqrt{2}$ to 2.

On the Average Number of Real Zeros of a Random Trigonometric Polynomial

This work emphasizes the special role played by semi-stable distribution which is the generalization of the stable distribution. Here \(a_0,~ a _1,\ldots ,a_n\) be a sequence of mutually independent random variables following semi-stable distribution with characteristic function \(exp \left( - \left( C + \cos {\log |t|} \right) |t|^{\alpha } \right) \), \(1 \le \alpha \le 2\) and \(C>1\) and \(b_1,~ b_2,\ldots ,b_n\) be positive constants. We then obtain the average number of zeros in the interval \([0, 2\pi ]\) of random trigonometric polynomial of the form \(T_n(\theta )=\sum \nolimits _{k=1}^{n}\left( \frac{a_0}{n}+a_kb_k\sin {k\theta }\right) \) for large n. The case when \(b_k=k^{\sigma -\frac{1}{\alpha }}\), \(\sigma =-\frac{2}{3\alpha }\) is studied in detail. Here this average is asymptotically equal to \(2n+o(1)\) except for a set of measure zero as \(n\rightarrow \infty \).

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 coefficients $(a_{k})_{k\ge 1}$ and $(b_{k})_{k\ge 1}$ are given by two independent stationary Gaussian processes with the same correlation function $\rho$. Under mild assumptions on the spectral function $\psi _\rho$ associated with $\rho$, we prove that the expectation of the number $N_n([0,2\pi ])$ of real roots of $f_n$ in the interval $[0,2\pi ]$ satisfies \[ \lim _{n \to +\infty } \frac {\mathbb E\left [N_n([0,2\pi ])\right ]}{n} = \frac {2}{\sqrt {3}}. \] The latter result not only covers the well-known situation of independent coefficients but allows us to deal with longrange correlations. In particular, it includes the case where the random coefficients are given by a fractional Brownian noise with any Hurst parameter.

Non universality for the variance of the number of real roots of random trigonometric polynomials

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 $$Y^i_{k}$$ , $$i\in \{1,2\}$$ , $$k\ge 1$$ , which are centered and standardized. We set $${\mathcal {N}}([0,\pi ],Y)$$ the number of real roots over $$[0,\pi ]$$ and $${\mathcal {N}}([0,\pi ],G)$$ 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 $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\text {Var}\left( {\mathcal {N}}_n([0,\pi ],Y)\right) }{n} =\lim _{n\rightarrow \infty }\frac{\text {Var}\left( {\mathcal {N}}_n([0,\pi ],G)\right) }{n} +\frac{1}{30}\left( {\mathbb {E}}\left( \left( Y_{1}^1\right) ^4\right) -3\right) . \end{aligned}$$ The latter establishes that the behavior of the variance is not universal and depends on the distribution of the underlying coefficients through their kurtosis. Actually, a more general result is proven in this article, which does not require that the coefficients are identically distributed. The proof mixes a recent result regarding Edgeworth expansions for distribution norms established in Bally et al. (Electron J Probab 23(45):1–51, 2018) with the celebrated Kac–Rice formula.

Universality of the nodal length of bivariate random trigonometric polynomials

Universality of the nodal length of bivariate random trigonometric polynomials

Expected number of real roots of random trigonometric polynomials

We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials Xn(t)=u+1n∑k=1n(Akcos(kt)+Bksin(kt)),t∈[0,2π],u∈R whose coefficients Ak,Bk, k∈N, are independent identically distributed random variables with zero mean and unit variance. If Nn[a,b] denotes the number of real roots of Xn in an interval [a,b]⊆[0,2π], we prove that limn→∞ENn[a,b]n=b−aπ3exp(−u22).

Local universality for real roots of random trigonometric polynomials

Consider a random trigonometric polynomial $X_n: \mathbb{R} \to \mathbb{R} $ of the form \[ X_n(t) = \sum _{k=1}^n \left ( \xi _k \sin (kt) + \eta _k \cos (kt)\right ), \] where $(\xi _1,\eta _1),(\xi _2,\eta _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 correlation function $(\sin t)/t$. We also establish similar local universality results for centered random vectors $(\xi _k,\eta _k)$ having an arbitrary covariance matrix or belonging to the domain of attraction of a two-dimensional $\alpha $-stable law.

CLT for the zeros of classical random trigonometric polynomials

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

Fluctuations of the Number of Critical Points of Random Trigonometric Polynomials

Abstract Denote by Znthe number of critical points of a random trigonometric polynomial of degree ≤ n We prove that as ν→∞ the expectation of Zν is asymptotic to 2ν √⅗ while its variance is asymptotic to δ∞νwhere δ∞≈ 0.35.

CLT for crossings of random trigonometric polynomials

We establish a central limit theorem for the number of roots of the equation $X_N(t) =u$ when $X_N(t)$ is a Gaussian trigonometric polynomial of degree $N$. The case $u=0$ was studied by Granville and Wigman. We show that for some size of the considered interval, the asymptotic behavior is different depending on whether $u$ vanishes or not. Our mains tools are: a) a chaining argument with the stationary Gaussain process with covariance $\sin(t)/t$, b) the use of Wiener chaos decomposition that explains some singularities that appear in the limit when $u \neq 0$.