Fr\'echet means of samples from a probability measure $\mu$ on any smoothly stratified metric space M with curvature bounded above are shown to satisfy a central limit theorem (CLT). The methods and results proceed by introducing and proving analytic properties of the "escape vector" of any finitely supported measure $\delta$ in M, which records infinitesimal variation of the Fr\'echet mean $\bar\mu$ of $\mu$ in response to perturbation of $\mu$ by adding the mass $t\delta$ for $t \to 0$. The CLT limiting distribution $N$ on the tangent cone $T$ at the Fr\'echet mean is characterized in four ways. The first uses tangential collapse $L$ to compare $T$ with a linear space and then applies a distortion map to the usual linear CLT to transfer back to $T$. Distortion is defined by applying escape after taking preimages under $L$. The second characterization constructs singular analogues of Gaussian measures on smoothly stratified spaces and expresses $N$ as the escape vector of any such "Gaussian mass". The third characterization expresses $N$ as the directional derivative, in the space of measures on $M$, of the barycenter map at $\mu$ in the (random) direction given by any Gaussian mass. The final characterization expresses $N$ as the directional derivative, in the space $C$ of continuous real-valued functions on $T$, of a minimizer map, with the derivative taken at the Fr\'echet function $F \in C$ along the (random) direction given by the negative of the Gaussian tangent field induced by $\mu$. Precise mild hypotheses on the measure $\mu$ guarantee these CLTs, whose convergence is proved via the second characterization of $N$ by formulating a duality between Gaussian masses and Gaussian tangent fields.

翻译：对于定义在曲率有上界的光滑分层度量空间$M$上的概率测度$\mu$，其样本的弗莱歇均值满足中心极限定理。本文通过引入并证明$M$上任意有限支撑测度$\delta$的"逃逸向量"的解析性质展开研究，该向量刻画了当扰动$t\delta$（$t \to 0$）叠加于$\mu$时，弗莱歇均值$\bar\mu$的无穷小变化。弗莱歇均值处切锥$T$上的CLT极限分布$N$通过四种方式表征。第一种方法利用切向坍缩$L$将$T$与线性空间进行比较，通过畸变映射将经典线性CLT转移回$T$，其中畸变定义为在$L$的原像上施加逃逸操作。第二种方法在光滑分层空间上构造高斯测度的奇异类比，并将$N$表达为任意此类"高斯质量"的逃逸向量。第三种方法将$N$表示为$M$上测度空间中重心映射在$\mu$处沿（随机）方向（由任意高斯质量给定）的方向导数。最后一种表征将$N$表示为$T$上连续实值函数空间$C$中极小化映射的方向导数，其中导数在弗莱歇函数$F \in C$处沿由$\mu$诱导的高斯切向量场负值所给定的（随机）方向计算。关于测度$\mu$的精确温和假设保证了这些CLT的有效性，其收敛性通过构建高斯质量与高斯切向量场之间的对偶性，借助$N$的第二种表征得以证明。