论文标题
光谱中心极限定理用于各向同性和固定高斯领域的添加功能
Spectral central limit theorem for additive functionals of isotropic and stationary Gaussian fields
论文作者
论文摘要
令$ b =（b_x）_ {x \ in \ mathbb {r}^d} $是$ n（0,1）$随机变量的集合，形成了$ \ mathbb {r}^d $上的真实价值的连续静态高斯字段，并设置$ c（x-y）= \ mathbb = $ c（x-y）令$φ：\ Mathbb {r} \ to \ Mathbb {r} $，以至于$ \ Mathbb {e} [φ（n）^2] <\ infty $带有$ n \ sim n（0,1）$，$ r $，让$ r $为$ y_t = $ y_t = \ int = \ td = \ int = \ int = \ int = $ t> 0 $，带有$ d \ subset \ mathbb {r}^d $ compact。 自从Breuer，Dobrushin，Major，Rosenblatt，Taqqu和其他人的80年代开创性的作品以来，以$ Y_T $的价格进行了中央和非中性限制定理，并且已不断地完善，扩展并应用于越来越多的不同情况，以至于它已成为其自身研究领域的领域。 普遍的信念代表了过去四十年中专家发展的直觉，是，作为$ t \ to \ infty $，其平均值的波动是$ y_t $的波动，通常（即在非常特殊的情况下除外），高斯时，当$ b $的记忆时间很短时，当$ b $ b $ b $ bugs and b $ bugs and b $ b $时，$ b $ b $ b $ nonth ynon nonth ynon n n $ b $。 我们在本文中表明，在过去的四十年中，这种直觉锻造可能是错误的，不仅是少数情况或在危重情况下。我们确实会发现各种情况下，$ y_t $在较长的内存环境中承认高斯波动。 为了实现这一目标，我们指出并证明了一个频谱中心限制定理，该定理将著名的Breuer-Major定理的结论扩展到了L^r（\ Mathbb {r}^d）$的情况。我们的主要数学工具是Malliavin-Stein方法和傅立叶分析技术。
Let $B=(B_x)_{x\in\mathbb{R}^d}$ be a collection of $N(0,1)$ random variables forming a real-valued continuous stationary Gaussian field on $\mathbb{R}^d$, and set $C(x-y)=\mathbb{E}[B_xB_y]$. Let $φ:\mathbb{R}\to\mathbb{R}$ be such that $\mathbb{E}[φ(N)^2]<\infty$ with $N\sim N(0,1)$, let $R$ be the Hermite rank of $φ$, and consider $Y_t = \int_{tD} φ(B_x)dx$, $t>0$, with $D\subset \mathbb{R}^d$ compact. Since the pioneering works from the 80s 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 $t\to\infty$ the fluctuations of $Y_t$ 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 $R\geq 2$. We show in this paper that this intuition forged over the last forty 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 $C\not\in L^R(\mathbb{R}^d)$. Our main mathematical tools are the Malliavin-Stein method and Fourier analysis techniques.
