Average Number of Real Zeros of Random Algebraic Polynomials Defined by the Increments of Fractional Brownian Motion

Mukeru, Safari

doi:10.1007/s10959-018-0818-0

Cited by 5 publications

(3 citation statements)

References 16 publications

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“…Consider its inverse matrix G −1 . A sequence of Gaussian random variables with covariance matrix G −1 shall be called the inverse fractional Gaussian noise of index h. Some properties of the zeros of the random polynomial ∑ n k=0 Δ k x k and the power series ∑ ∞ k=0 Δ n x n , where (Δ n ) is the fractional Gaussian noise are given in [11] and [13]. Here, we are interested in the function f (z) = ∑ ∞ n=1 n z n−1 , where ( n ) is the inverse fractional Gaussian noise.…”

Section: Inverse Fractional Gaussian Noisementioning

confidence: 99%

Zeros of Gaussian power series, Hardy spaces and determinantal point processes

Mukeru

Mulaudzi

2021

Ann. Funct. Anal.

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Given a sequence ( n ) of standard i.i.d complex Gaussian random variables, Peres and Virág (in the paper "Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process" Acta Math. (2005) 194, 1-35) discovered the striking fact that the zeros of the random power series f (z) = ∑ ∞ n=1 n z n−1 in the complex unit disc constitute a determinantal point process. The study of the zeros of the general random series f(z), where the restriction of independence is relaxed upon the random variables ( n ) is an important open problem. This paper proves that if ( n ) is an infinite sequence of complex Gaussian random variables, such that their covariance matrix is invertible and its inverse is a Toeplitz matrix, then the zero set of f(z) constitutes a determinantal point process with the same distribution as the case of i.i.d variables studied by Peres and Virág. The arguments are based on some interplays between Hardy spaces and reproducing kernels. Illustrative examples are constructed from classical Toeplitz matrices and the classical fractional Gaussian noise.

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Section: Inverse Fractional Gaussian Noisementioning

confidence: 99%

Zeros of Gaussian power series, Hardy spaces and determinantal point processes

Mukeru

Mulaudzi

2021

Ann. Funct. Anal.

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“…Consider its inverse matrix G −1 . A sequence of Gaussian random variables with covariance matrix G −1 shall be called the inverse fractional Gaussian noise of index h. Some properties of the zeros of the random polynomial n k=0 ∆ k x k and the power series ∞ k=0 ∆ n x n where (∆ n ) is the fractional Gaussian noise are given in [11] and [13]. Here we are interested in the function f (z) = ∞ n=1 ξ n z n−1 where (ξ n ) is the inverse fractional Gaussian noise.…”

Section: Inverse Fractional Gaussian Noisementioning

confidence: 99%

Zeros of Gaussian power series, Hardy spaces and determinantal point processes

Mukeru¹,

Mulaudzi²

2021

Preprint

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Given a sequence (ξ n ) of standard i.i.d complex Gaussian random variables, Peres and Virág (in the paper "Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process" Acta Math. (2005) 194, 1-35) discovered the striking fact that the zeros of the random power series f (z) = ∞ n=1 ξ n z n−1 in the complex unit disc D constitute a determinantal point process. The study of the zeros of the general random series f (z) where the restriction of independence is relaxed upon the random variables (ξ n ) is an important open problem. This paper proves that if (ξ n ) is an infinite sequence of complex Gaussian random variables such that their covariance matrix is invertible and its inverse is a Toeplitz matrix, then the zero set of f (z) constitutes a determinantal point process with the same distribution as the case of i.i.d variables studied by Peres and Virág. The arguments are based on some interplays between Hardy spaces and reproducing kernels. Illustrative examples are constructed from classical Toeplitz matrices and the classical fractional Gaussian noise.

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“…Recently, the universality of these asymptotics has been established in a certain number of models, see e.g. [Kac43,IM68,Far86,Mat10,Muk18,NNV15,DNV18] in the case of algebraic polynomials and [AP15, ADL, Fla17, IKM16, ADP19] in the case of trigonometric polynomials. The notion of universality stands here for the fact that these asymptotics do not depend on the choice of the law of the random entries, and to a certain extent, nor their correlation.…”

Section: Real Zeros Of Random Trigonometric Polynomialsmentioning

confidence: 99%

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

Pautrel¹

2020

Electron. Commun. Probab.

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We consider random trigonometric polynomials of the formwhere (a k ) k≥1 and (b k ) k≥1 are two independent stationary Gaussian processes with the same correlation function ρ : k → cos(kα), with α ≥ 0. We show that the asymptotics of the expected number of real zeros differ from the universal one 2 √ 3 , holding in the case of independent or weakly dependent coefficients. More precisely, for all ε > 0, for all ∈ ( √ 2, 2], there exists α ≥ 0 and n ≥ 1 large enough such thatwhere N (fn, [0, 2π]) denotes the number of real zeros of the function fn in the interval [0, 2π]. Therefore, this result provides the first example where the expected number of real zeros does not converge as n goes to infinity by exhibiting a whole range of possible subsequential limits ranging from √ 2 to 2.

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Average Number of Real Zeros of Random Algebraic Polynomials Defined by the Increments of Fractional Brownian Motion

Cited by 5 publications

References 16 publications

Zeros of Gaussian power series, Hardy spaces and determinantal point processes

Zeros of Gaussian power series, Hardy spaces and determinantal point processes

Zeros of Gaussian power series, Hardy spaces and determinantal point processes

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

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