Rates of convergence for the number of zeros of random trigonometric polynomials

TL;DR

This paper investigates how quickly the distribution of zeros of random trigonometric polynomials converges to that of a Gaussian process, using Wasserstein-1 distance and Kac-Rice formula for approximation.

Contribution

It establishes the rate of convergence of the number of zeros of RTPs to a Gaussian process, leveraging Donsker's theorem and approximation techniques.

Findings

Convergence in Wasserstein-1 distance established

Rate of convergence for the number of zeros derived

Use of Kac-Rice formula for approximation of zero count

Abstract

In this paper, we quantify the rate of convergence between the distribution of number of zeros of random trigonometric polynomials (RTP) with i.i.d. centered random coefficients and the number of zeros of a stationary centered Gaussian process G, whose covariance function is given by the sinc function. First, we find the convergence of the RTP towards G in the Wasserstein-1 distance, which in turn is a consequence of Donsker Theorem. Then, we use this result to derive the rate of convergence between their respective number of zeros. Since the number of real zeros of the RTP is not a continuous function, we use the Kac-Rice formula to express it as the limit of an integral and, in this way, we approximate it by locally Lipschitz continuous functions.