Electronic Journal of Probability

Local universality for real roots of random trigonometric polynomials

Alexander Iksanov, Zakhar Kabluchko, and Alexander Marynych

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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.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 63, 19 pp.

Received: 1 June 2016
Accepted: 5 October 2016
First available in Project Euclid: 17 October 2016

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Zentralblatt MATH identifier

Primary: 26C10: Polynomials: location of zeros [See also 12D10, 30C15, 65H05]
Secondary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} 42A05: Trigonometric polynomials, inequalities, extremal problems 60F17: Functional limit theorems; invariance principles 60G55: Point processes

random trigonometric polynomials real zeros local universality stationary processes random analytic functions functional limit theorem stable processes

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Iksanov, Alexander; Kabluchko, Zakhar; Marynych, Alexander. Local universality for real roots of random trigonometric polynomials. Electron. J. Probab. 21 (2016), paper no. 63, 19 pp. doi:10.1214/16-EJP9. https://projecteuclid.org/euclid.ejp/1476706888

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