Poisson-Sampled Fr\'echet Means on Gaussian Information Manifolds

Abstract

Distribution-valued marks in a spatial network live on a statistical manifold, whereas their sampling locations are governed by stochastic geometry. We develop a rigorous finite-window theory for Fr\'echet means sampled by a Poisson point process. The population target is the barycenter of the normalized window mark law, avoiding the typically divergent unnormalized objective on the whole space. Conditional on the Poisson count, independently marked points form an ordinary independent sample; this yields an exact zero-truncated Poissonization transform for fixed-sample risks, tails, consistency, and intrinsic central limit theorems. For geodesic-supported marks, we obtain closed-form random-count mean-square errors. For a common spatial random field, the error separates exactly into a correlation floor and a Poisson term, and stationary Gaussian fields satisfy a spatial central limit theorem with variance equal to the integrated field covariance plus a Poisson diagonal contribution. Slivnyak's theorem and Poisson splitting then give valid reduced- and non-reduced Palm laws and exact resampling and thinning formulas; the correlation floor cancels when the retained and full barycenters share the same field realization. The theory is specialized to covariance-varying Gaussian models: the full univariate Fisher--Rao manifold and the affine-invariant manifold of multivariate covariance matrices, including arbitrary noncommuting, spatially heterogeneous covariance laws. The Wasserstein discussion records the correct one-dimensional covariance barycenter and isolates its commuting flat boundary case.