SPECTRAL CENTRAL LIMIT THEOREM FOR ADDITIVE FUNCTIONALS OF ISOTROPIC AND STATIONARY GAUSSIAN FIELDS

• 成果类型: Article
• 署名作者: Maini, Leonardo; Nourdin, Ivan
• 署名单位: University of Luxembourg
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/23-AOP1669
• 发表日期: 2024
• 页码: 737-763
• 关键词: nonlinear functionals

摘要

Let B = (Bx)(x is an element of Rd) be a collection of N(0, 1) random variables forming a real -valued continuous stationary Gaussian field on R-d, and set C(x -y) = E[BxBy]. Let yo : R -> R be such that E[phi(N)(2)] < infinity with N similar to N(0, 1), let R be the Hermite rank of yo, and consider Y-t = integral(tD) phi(B-x)dx, t > 0 with D subset of R-d compact. Since the pioneering works from the 1980s by Breuer, Dobrushin, Major, Rosenblatt, Taqqu and others, central and noncentral limit theorems for Y-t have been constantly refined, extended and applied to an increasing number of diverse situations, to such an extent that it has become a field of research in its own right. The common belief, representing the intuition that specialists in the subject have developed over the last four decades, is that as t -> infinity the fluctuations of Yt around its mean are, in general (i.e., except possibly in very special cases), Gaussian when B has short memory, and non -Gaussian when B has long memory and R >= 2. We show in this paper that this intuition forged over the last 40 years can be wrong, and not only marginally or in critical cases. We will indeed bring to light a variety of situations where Y-t admits Gaussian fluctuations in a long memory context. To achieve this goal, we state and prove a spectral central limit theorem, which extends the conclusion of the celebrated Breuer-Major theorem to situations where C is an element of L (R)(Rd). Our main mathematical tools are the Malliavin- Stein method and Fourier analysis techniques.