A QUANTITATIVE CENTRAL LIMIT THEOREM FOR THE EULER-POINCARE CHARACTERISTIC OF RANDOM SPHERICAL EIGENFUNCTIONS

• 成果类型: Article
• 署名作者: Cammarota, Valentina; Marinucci, Domenico
• 署名单位: Sapienza University Rome; University of Rome Tor Vergata
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1245
• 发表日期: 2018
• 页码: 3188-3228
• 关键词: arithmetic random waves; gaussian random-fields; excursion probability; harmonics; fluctuations; number

摘要

We establish here a quantitative central limit theorem (in Wasserstein distance) for the Euler-Poincare characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler-Poincare characteristic into different Wiener-chaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level u is fully degenerate, that is, the Euler-Poincare characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. We discuss also a possible unifying framework for the Lipschitz-Killing curvatures of the excursion sets for Gaussian spherical harmonics.