论文标题
通过扩散流形及其应用来构建可交换对
Constructing exchangeable pairs by diffusion on manifolds and its application
论文作者
论文摘要
我们通过通过歧管上的扩散过程扰动随机变量来构建一个连续的可交换对家族，以便将Stein方法应用于某些几何设置。我们通过扩散方法将我们的扰动与构建可交换对的其他方法进行比较，并表明我们的扰动方案与Stein的无限版本和和谐的方法合作。更确切地说，我们的可交换对满足了一般的无限Stein方法中的关键条件。基于可交换对，我们能够扩展Laplacian在紧凑型歧管上的特征函数的近似正态性到Witten Laplacian的特征函数，即形式：$Δ_W=Δ_w=δ-\ \ \ nabla h $。然后，我们应用摘要定理以恢复球体线性统计数据的中心极限结果。最后，我们证明了Stein的指数分配方法的无限版本，并将其与我们连续的可交换对系列结合使用，以扩展$ | tr u |^2 $的近似指数结果，其中$ tr u $是从矩阵$ u $的第一个力量的痕迹中，从矩阵$ u $中取得了单位群体，以使其构图且通用量子的量子及其通用量子。
We construct a continuous family of exchangeable pairs by perturbing the random variable through diffusion processes on manifold in order to apply Stein method to certain geometric settings. We compare our perturbation by diffusion method with other approaches of building exchangeable pairs and show that our perturbation scheme cooperates with the infinitesimal version of Stein's method harmoniously. More precisely, our exchangeable pairs satisfy a key condition in the infinitesimal Stein's method in general. Based on the exchangeable pairs, we are able to extend the approximate normality of eigenfunctions of Laplacian on compact manifold to eigenfunctions of Witten Laplacian, which is of the form:$Δ_w = Δ- \nabla H$. We then apply our abstract theorem to recover a central limit result of linear statistics on sphere. Finally, we prove an an infinitesimal version of Stein's method for exponential distribution and combine it with our continuous family of exchangeable pairs to extend an approximate exponentiality result of $|Tr U|^2$, where $Tr U$ is the trace of the first power of a matrix $U$ sampled from the Haar measure of unitary group, to arbitrary power and its analog for general circular ensemble.
