Central limit theorems for Fr\'echet means on stratified spaces

Abstract

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.