Random trigonometric polynomials: universality and non-universality of the variance for the number of real roots

TL;DR

This paper investigates the variance of the number of real roots in random trigonometric polynomials with iid coefficients, revealing dependence on kurtosis and contrasting with classical models like Kac polynomials.

Contribution

It establishes the asymptotic linear variance dependence on kurtosis for general coefficient distributions, including discrete ones, extending beyond classical Kac polynomial results.

Findings

Variance is asymptotically linear in expectation, with constant depending on kurtosis.

Results apply to general coefficient distributions, including discrete cases.

Introduces a comparison framework using Edgeworth expansion and characteristic functions.

Abstract

In this paper, we study the number of real roots of random trigonometric polynomials with iid coefficients. When the coefficients have zero mean, unit variance and some finite high moments, we show that the variance of the number of real roots is asymptotically linear in terms of the expectation; furthermore, the multiplicative constant in this linear relationship depends only on the kurtosis of the common distribution of the polynomial's coefficients. This result is in sharp contrast to the classical Kac polynomials whose corresponding variance depends only on the first two moments. Our result is perhaps the first paper to establish the variance for general distribution of the coefficients including discrete ones, for a model of random polynomials outside the family of the Kac polynomials. Our method gives a fine comparison framework throughout Edgeworth expansion, asymptotic Kac-Rice formula and a detailed analysis of characteristic functions.