In this article, we consider the following family of random trigonometric polynomials $p_n(t,Y)=sum_{k=1}^n Y_{k}^1 cos(kt)+Y_{k}^2sin(kt)$ for a given sequence of i.i.d. random variables %${Y_{k,1},Y_{k,2}}_{kge 1}$ $Y^i_{k}$, $iin{1,2}$, $kge 1$, 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 egin{equation*} lim_{n oinfty}rac{ ext{Var}left(mathcal{N}_n([0,pi],Y) ight)}{n}% =lim_{n oinfty}rac{ ext{Var}left(mathcal{N}_n([0,pi],G) ight)}{n}% +rac{1}{30}left(mathbb{E}((Y_{1}^1)^4)-3 ight). end{equation*} 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.

Bally, V., Caramellino, L., Poly, G. (2019). Non universality for the variance of the number of real roots of random trigonometric polynomials. PROBABILITY THEORY AND RELATED FIELDS, 174(3-4), 887-927 [10.1007/s00440-018-0869-2].

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

Caramellino, L;
2019-01-01

Abstract

In this article, we consider the following family of random trigonometric polynomials $p_n(t,Y)=sum_{k=1}^n Y_{k}^1 cos(kt)+Y_{k}^2sin(kt)$ for a given sequence of i.i.d. random variables %${Y_{k,1},Y_{k,2}}_{kge 1}$ $Y^i_{k}$, $iin{1,2}$, $kge 1$, 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 egin{equation*} lim_{n oinfty}rac{ ext{Var}left(mathcal{N}_n([0,pi],Y) ight)}{n}% =lim_{n oinfty}rac{ ext{Var}left(mathcal{N}_n([0,pi],G) ight)}{n}% +rac{1}{30}left(mathbb{E}((Y_{1}^1)^4)-3 ight). end{equation*} 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.
2019
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/06 - PROBABILITA' E STATISTICA MATEMATICA
English
Random trigonometric polynomials; Edgeworth expansion for non smooth functions; Kac–Rice formula; Small balls estimates
http://www.mat.uniroma2.it/~caramell/pubblVQR2015-19/1.PTRF2019.pdf
Bally, V., Caramellino, L., Poly, G. (2019). Non universality for the variance of the number of real roots of random trigonometric polynomials. PROBABILITY THEORY AND RELATED FIELDS, 174(3-4), 887-927 [10.1007/s00440-018-0869-2].
Bally, V; Caramellino, L; Poly, G
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/207666
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