[Paper Review] CLT for the zeros of Classical Random Trigonometric Polynomials

Jean‐Marc Azäis, Federico Dalmao+1 more|arXiv (Cornell University)|Jan 22, 2014

Geometry and complex manifolds20 references22 citations

TL;DR

This paper establishes a Central Limit Theorem (CLT) for the number of zeros of classical random trigonometric polynomials of the form $ K^{-1/2} \sum_{n=1}^K a_n \cos(nt) $, where $ a_n $ are i.i.d. standard Gaussian variables. Using Hermite/Wiener chaos decomposition and Rice's formula for zero counting, the authors prove that the normalized number of zeros on $[0, \pi]$ converges in distribution to a normal random variable with variance $ V^2 $, where $ V^2 \approx 0.089 $. The result confirms the asymptotic variance behavior conjectured by Farahmand and Granville-Wigman, with a simplified proof compared to prior work.

ABSTRACT

We prove a Central Limit Theorem for the number of zeros of random trigonometric polynomials of the form $K^{-1/2}\sum_{n=1}^{K} a_n\cos(nt)$, being $(a_n)_n$ independent standard Gaussian random variables. In particular, we prove the conjecture by Farahmand, Granville & Wigman that the variance is equivalent to $V^2K$, $0<V^2<\infty$, as $K o\infty$. % The case of stationary trigonometric polynomials was studied by Granville & Wigman and by Aza\"\is & Le\'on. Our approach is based on the Hermite/Wiener-Chaos decomposition for square-integrable functionals of a Gaussian process and on Rice Formula for zero counting.

Motivation & Objective

• To establish a Central Limit Theorem (CLT) for the number of real zeros of classical random trigonometric polynomials on $[0, \pi]$.
• To provide a simplified proof of the asymptotic variance behavior $ \sim V^2 K\pi $, previously conjectured and proven by Su & Shao.
• To extend the CLT to non-stationary Gaussian processes via a limiting argument based on Wiener chaos decomposition.
• To demonstrate that the limit variance for the non-stationary case matches that of the stationary counterpart $ X_K(t) $, despite different covariance structures.
• To avoid reliance on high-order moment conditions by using Peccati-Tudor's method and Hermite expansions.

Proposed method

• Apply the Wiener-Itô chaos decomposition to the normalized number of zeros of $ \tilde{T}_K(t) = K^{-1/2} \sum_{n=1}^K a_n \cos(nt/K) $ on $[0, K\pi]$.
• Use Rice's formula to express the expected number of zeros as an integral involving the density and conditional expectation of the derivative given the process is zero.
• Analyze the covariance structure of $ \tilde{T}_K $ and show its behavior near zero is asymptotically equivalent to that of the stationary process $ \tilde{X}_K $, enabling variance comparison.
• Leverage the Peccati-Tudor criterion for CLT in Wiener chaos to establish convergence to a normal distribution without requiring higher-than-second-order moment conditions.
• Use uniform bounds on the covariance and its derivatives via Taylor expansions and Dudley's theorem to control the sup-norm of high-order derivatives.
• Apply Arcones' inequality and Hermite expansion techniques to control off-diagonal terms in the second moment of the zero-counting functional.

Experimental results

Research questions

• RQ1 Does the number of zeros of the classical random trigonometric polynomial $ T_K(t) = K^{-1/2} \sum_{n=1}^K a_n \cos(nt) $ satisfy a Central Limit Theorem as $ K \to \infty $?
• RQ2 What is the asymptotic variance of the number of zeros of $ T_K $ on $[0, \pi]$, and does it match the conjectured form $ V^2 K\pi $?
• RQ3 Can the CLT for the zero count be established without imposing moment conditions beyond the second order?
• RQ4 How does the non-stationary structure of $ T_K $ affect the zero-counting distribution compared to the stationary $ X_K $?
• RQ5 Is the limit variance of the zero count for $ T_K $ the same as for the stationary process $ X_K $?

Key findings

• The normalized number of zeros of $ T_K $ on $[0, \pi]$, scaled by $ \sqrt{\pi K} $, converges in distribution to a normal random variable with mean zero and variance $ V^2 $, where $ 0 < V^2 < \infty $.
• The asymptotic variance of the number of zeros is $ \sim V^2 K\pi $, with $ V^2 \approx 0.089 $, confirming the conjecture by Farahmand & Sambandham and Granville & Wigman.
• The limit variance for $ T_K $ matches that of the stationary trigonometric polynomial $ X_K $, despite different covariance structures, due to similar local behavior near zero.
• The CLT is established via the Peccati-Tudor method in Wiener chaos, avoiding the need for higher-order moment conditions used in prior works.
• Numerical simulations confirm the derived value of $ V^2 \approx 0.089 $, contradicting the value $ c \approx 0.257 $ reported by Su & Shao for the same variance, suggesting a discrepancy in that result.
• A CLT is also established for the number of zeros of a non-stationary Gaussian process $ T $ with covariance $ r(s,t) = \frac{1}{2}(\text{sc}(t-s) + \text{sc}(t+s)) $, showing convergence to standard normal after standardization.