## Electronic Journal of Probability

### Local universality for real roots of random trigonometric polynomials

#### Abstract

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

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

Dates
Accepted: 5 October 2016
First available in Project Euclid: 17 October 2016

https://projecteuclid.org/euclid.ejp/1476706888

Digital Object Identifier
doi:10.1214/16-EJP9

Mathematical Reviews number (MathSciNet)
MR3563891

Zentralblatt MATH identifier
1361.30009

#### Citation

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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