[Paper Review] Non universality for the variance of the number of real roots of random trigonometric polynomials

Vlad Bally, Lucia Caramellino+1 more|arXiv (Cornell University)|Nov 9, 2017

Geometry and complex manifolds19 references21 citations

TL;DR

This paper establishes non-universality in the variance of the number of real roots of random trigonometric polynomials by showing that the asymptotic variance depends on the kurtosis of the coefficient distribution. Using the Kac-Rice formula and Edgeworth expansions, it proves that the variance grows linearly with n, with a correction term proportional to the excess kurtosis, specifically $\frac{1}{30}(\mathbb{E}(Y_{1,1}^4) - 3)$, breaking universality seen in Kac polynomials.

ABSTRACT

In this article, we consider the following family of random trigonometric polynomials $p_n(t,Y)=\\sum_{k=1}^n Y_{k,1} \\cos(kt)+Y_{k,2}\\sin(kt)$ for a given sequence of i.i.d. random variables $\\{Y_{k,1},Y_{k,2}\\}_{k\\ge 1}$ 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 $$ \\lim_{n\ o\\infty}\\frac{\ ext{Var}\\left(\\mathcal{N}_n([0,\\pi],Y)\ ight)}{n} =\\lim_{n\ o\\infty}\\frac{\ ext{Var}\\left(\\mathcal{N}_n([0,\\pi],G)\ ight)}{n} +\\frac{1}{30}\\left(\\mathbb{E}(Y_{1,1}^4)-3\ ight). $$ 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's expansions for distribution norms established in arXiv:1606.01629 with the celebrated Kac-Rice formula.

Motivation & Objective

• To investigate whether the variance of the number of real roots in random trigonometric polynomials is universal across coefficient distributions.
• To determine if the asymptotic variance depends on higher moments of the coefficients, particularly kurtosis.
• To extend universality results—previously known for Kac polynomials—to trigonometric polynomials, revealing a key difference in behavior.
• To establish a precise correction term for the variance that depends on the fourth moment of the coefficients.

Proposed method

• The authors use the Kac-Rice formula to express the variance of the number of real roots as an integral over the joint density of the process and its derivative.
• They apply a recent result on Edgeworth expansions for non-smooth functionals of weakly dependent random vectors to analyze the distribution of the root count.
• A scaling transformation is introduced to map the original trigonometric polynomial to a process converging to a stationary Gaussian process with sinc correlation.
• The proof relies on Doeblin's condition and moment conditions to ensure weak dependence and convergence in distribution.
• The covariance structure of the limiting Gaussian process is analyzed to derive the asymptotic variance, with a focus on the role of the fourth cumulant.
• Small ball estimates and spectral analysis of the covariance matrix are used to control the behavior of the root count in small intervals.

Experimental results

Research questions

• RQ1 Does the asymptotic variance of the number of real roots in random trigonometric polynomials depend on the distribution of the coefficients beyond their second moment?
• RQ2 Can a non-universal correction term be derived for the variance, and if so, what is its dependence on the kurtosis of the coefficients?
• RQ3 How does the behavior of the variance in trigonometric polynomials differ from that in Kac polynomials, where universality is known to hold?
• RQ4 To what extent can Edgeworth expansions be applied to non-smooth functionals of dependent random vectors in this context?
• RQ5 Is the limiting behavior of the root count distribution universal when higher moments of coefficients are varied?

Key findings

• The asymptotic variance of the number of real roots grows linearly with n, with a correction term proportional to the excess kurtosis of the coefficients.
• The key result is that $\lim_{n\to\infty}\frac{\text{Var}(\mathcal{N}_n([0,\pi],Y))}{n} = \lim_{n\to\infty}\frac{\text{Var}(\mathcal{N}_n([0,\pi],G))}{n} + \frac{1}{30}(\mathbb{E}(Y_{1,1}^4) - 3)$, proving non-universality.
• The correction term depends only on the fourth moment of the coefficients, specifically on their kurtosis, showing that the variance is not universal across distributions.
• The result holds under Doeblin's condition and moment conditions, without requiring identical distribution of coefficients.
• The proof technique combines the Kac-Rice formula with a novel application of Edgeworth expansions for non-smooth functionals.
• The analysis confirms that the limiting Gaussian process has a well-defined covariance structure that captures the distributional dependence on the fourth cumulant.