论文标题
紧凑型公制空间上的中央限制定理
Central Limit Theorems on Compact Metric Spaces
论文作者
论文摘要
我们生产一系列与紧凑的度量度量空间$（k，d，η）$相关的中央限制定理（CLT），并具有合理的概率度量。对于第一个CLT，我们可以通过将$ k $嵌入$ {\ Mathcal C}（k）$的$η$忽略$η$，这是$ k $与SUP Norm的连续函数的空间，然后在Banach Space上（Theorem 3.1）将已知的CLT用于样品均值。但是，示例平均值在$ k $上没有意义，因此使用$η$，我们为样本fréchet平均值开发了CLT（推论4.1）。这涉及在$ k $的嵌入式图像的封闭凸壳上工作。要在$ l^2（k，η）$的更轻松的希尔伯特空间设置中工作，我们必须将公制$ d $修改为相关的公制$d_η$。我们获得了$ l^2 $ -clt的样本平均值和样本fréchet平均值（定理5.1），并且我们将封闭凸面上的fréchet样品和种群平均值与$ k $的图像相关联。由于$ l^2 $和$ l^\ infty $ norms扮演着重要角色，因此在第6节中，我们开发了与$ d $和$η$相关的度量量标准，所有$ l^p $ norms均等效。
We produce a series of Central Limit Theorems (CLTs) associated to compact metric measure spaces $(K,d,η)$, with $η$ a reasonable probability measure. For the first CLT, we can ignore $η$ by isometrically embedding $K$ into ${\mathcal C}(K)$, the space of continuous functions on $K$ with the sup norm, and then applying known CLTs for sample means on Banach spaces (Theorem 3.1). However, the sample mean makes no sense back on $K$, so using $η$ we develop a CLT for the sample Fréchet mean (Corollary 4.1). This involves working on the closed convex hull of the embedded image of $K$. To work in the easier Hilbert space setting of $L^2(K,η)$, we have to modify the metric $d$ to a related metric $d_η$. We obtain an $L^2$-CLT for both the sample mean and the sample Fréchet mean (Theorem 5.1), and we relate the Fréchet sample and population means on the closed convex hull to the Fréchet means on the image of $K$. Since the $L^2$ and $L^\infty$ norms play important roles, in Section 6 we develop a metric-measure criterion relating $d$ and $η$ under which all $L^p$ norms are equivalent.
