Zeroes of Gaussian analytic functions with translation-invariant distribution

Feldheim, Naomi

doi:10.1007/s11856-012-0130-0

Cited by 16 publications

(27 citation statements)

References 19 publications

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“…They are thus less amenable to traditional statistical signal processing procedures, which often assume some stationarity. However, zeros of symmetric GAFs are usually well approximated by the zeros of GAFs, see [Prosen, 1996, Feldheim, 2013. We refer the reader to [Bardenet et al, 2017, Section 4.3] for another discussion on the Gabor transform of a real white noise and its approximation by that of a proper complex white noise.…”

Section: More Transforms More Gafs and Related Resultsmentioning

confidence: 99%

Time-frequency transforms of white noises and Gaussian analytic functions

Bardenet

Hardy

2021

Applied and Computational Harmonic Analysis

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A family of Gaussian analytic functions (GAFs) has recently been linked to the Gabor transform of white Gaussian noise [Bardenet et al., 2017]. This answered pioneering work by Flandrin [2015], who observed that the zeros of the Gabor transform of white noise had a very regular distribution and proposed filtering algorithms based on the zeros of a spectrogram. The mathematical link with GAFs provides a wealth of probabilistic results to inform the design of such signal processing procedures. In this paper, we study in a systematic way the link between GAFs and a class of time-frequency transforms of Gaussian white noises on Hilbert spaces of signals. Our main observation is a conceptual correspondence between pairs (transform, GAF) and generating functions for classical orthogonal polynomials. This correspondence covers some classical time-frequency transforms, such as the Gabor transform and the Daubechies-Paul analytic wavelet transform. It also unveils new windowed discrete Fourier transforms, which map white noises to fundamental GAFs. All these transforms may thus be of interest to the research program "filtering with zeros". We also identify the GAF whose zeros are the extrema of the Gabor transform of the white noise and derive their first intensity. Moreover, we discuss important subtleties in defining a white noise and its transform on infinite dimensional Hilbert spaces. Finally, we provide quantitative estimates concerning the finite-dimensional approximations of these white noises, which is of practical interest when it comes to implementing signal processing algorithms based on GAFs.

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Section: More Transforms More Gafs and Related Resultsmentioning

confidence: 99%

Time-frequency transforms of white noises and Gaussian analytic functions

Bardenet

Hardy

2021

Applied and Computational Harmonic Analysis

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“…where µ f (·) is some absolutely continuous, non-negative measure on R (see [7,Theorem 1]). Further, z → f (z) is stationary under real translations, hence for any x j ∈ R and r ∈ [δe β , 1 2…”

Section: 2mentioning

confidence: 99%

Exponential Concentration for Zeroes of Stationary Gaussian Processes

Basu

Dembo

Feldheim

et al. 2018

International Mathematics Research Notices

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We show that for any centered stationary Gaussian process of integrable covariance, whose spectral measure has compact support, or finite exponential moments (and some additional regularity), the number of zeroes of the process in [0, T ] is within ηT of its mean value, up to an exponentially small in T probability. IntroductionThe study of zeroes of Gaussian stationary processes goes back at least to Kac [10] and Rice [19]. Since then, much work on this topic appeared in the statistics, physics, mathematics and engineering literature. One of the earliest and most fundamental results in this area is the Kac-Rice formula, which calculates the mean number of zeroes in any interval. A similar formula may be written for the variance of the number of zeroes, but it is much harder to analyze. It was only many years after Kac and Rice that fluctuations and central limit theorems were better understood, with works by Cuzick [5], Slud [21], Azaïs-León [2] and others. Questions of large deviations, that is, estimation of the rare event of having many more or much less zeroes than expected in a long interval, remained almost unexplored. One particular such event is that of having no zeroes at all in a long interval, which is also known by the name of "persistence". Results (and speculations) about this event were initiated by Slepian [20] and were better understood only recently [8].In the meantime, complex zeroes of certain Gaussian analytic functions received much attention. Most notably, zeroes of the Fock-Bargmann model were introduced by Sodin-Tsirelson [23] and extensively studied by many authors since then. This model has a remarkable property: its zeroes form a point process in the plane with quadratic repulsion, and invariance of distribution under all planar isometries. Sodin-Tsirelson proved the asymptotic normality of these zeroes in [23], and moreover, an exponential concentration of the zeroes around the mean in [24] (See also [12,17] for more about concentration, and [16] for finer results on asymptotic normality). Exponential concentration was proved for other related models, such as nodal lines of spherical harmonics in [15]. Inspired by these works, the question of concentration for real zeroes got some attention [22], [26, Thm. 2c1], but, until now, was not settled even for a single particular example.The aim of this paper is to prove exponential concentration for real zeroes of certain Gaussian stationary functions on R, which have an analytic extension to a strip in the complex plane and smooth spectral density. These conditions allow us to use tools from complex analysis, thus generalizing the mechanics of the aforementioned works on the Fock-Bargmann model.We consider here centered Gaussian stationary processes {X(t) : t ∈ R} having a.s. absolutely continuous sample path. That is, random absolutely continuous functions f : R → R, whose finite

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“…They were also able to obtain a limit of the intensity function as n → ∞. Generalizations to other types of real-valued random variables and to other random polynomials with basis functions different than the monomials were made by Ibragimov and Zeitouni [20], Feildheim [17], and Vanderbei [38].…”

Section: Introductionmentioning

confidence: 99%

“…, n, and all z ∈ C. Let N n (Ω) denote the (random) number of zeros of P n (z) as defined by (1) in a Jordan region Ω of the complex plane. Due to Edelman and Kostlan [8] (with different proofs later given by Hough, Krishnapur, Peres, and Virág in [19], Feldheim [17], the author [42], and Ledoan [23]) it is known that for each Jordan region Ω ⊂ {z ∈ C : K n (z, z) = 0}, we have that the intensity function ρ n associated to P n satisfies…”

Section: Introductionmentioning

confidence: 99%

Zeros of random orthogonal polynomials with complex Gaussian coefficients

Yeager¹

2018

Rocky Mountain J. Math.

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Let {f j } n j=0 be a sequence of orthonormal polynomials where the orthogonality relation is satisfied on either the real line or on the unit circle. We study zero distribution of random linear combinations of the formwhere η 0 , . . . , η n are complex-valued i.i.d. standard Gaussian random variables. Using the Christoffel-Darboux formula, the density function for the expected number of zeros of P n in these cases takes a very simple shape. From these expressions, under the mere assumption that the orthogonal polynomials are from the Nevai class, we give the limiting value of the density function away from their respective sets where the orthogonality holds. In the case when {f j } are orthogonal polynomials on the unit circle, the density function shows that the expected number of zeros of P n are clustering near the unit circle. To quantify this phenomenon, we give a result that estimates the expected number of complex zeros of P n in shrinking neighborhoods of compact subsets of the unit circle.

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Zeroes of Gaussian analytic functions with translation-invariant distribution

Cited by 16 publications

References 19 publications

Time-frequency transforms of white noises and Gaussian analytic functions

Time-frequency transforms of white noises and Gaussian analytic functions

Exponential Concentration for Zeroes of Stationary Gaussian Processes

Zeros of random orthogonal polynomials with complex Gaussian coefficients

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