2023/11/15 by Mattingly, Jonathan C., Miller, Ezra, Tran, Do · 1 citation
#28C99 #47H14 #49J50 #49J52 #53C23 #53C80 #57N80 #58A35 #58C20 #58Z05 #60B05 #60D05 #62G20 #62G35 #62R07 #62R20 #62R30 #92B10 #FOS: Mathematics #Metric Geometry (math.MG) #Primary: 60F05 #Probability (math.PR) #Secondary: 58K30 #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2311.09455
Fréchet means of samples from a probability measure μ 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 δ in M, which records infinitesimal variation of the Fréchet mean μ of μ in response to perturbation of μ by adding the mass tδ for t → 0. The CLT limiting distribution N on the tangent cone T at the Fréchet 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 μ 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 Fréchet function F ∈ C along the (random) direction given by the negative of the Gaussian tangent field induced by μ. Precise mild hypotheses on the measure μ guarantee these CLTs, whose convergence is proved via the second characterization of N by formulating a duality between Gaussian masses and Gaussian tangent fields.