Fréchet Means in Infinite Dimensions

Abstract

While there exists a well-developed asymptotic theory of Fréchet means of random variables taking values in a general "finite-dimensional" metric space, there are only a few known results in which the random variables can take values in an "infinite-dimensional" metric space. Presently, we develop a general asymptotic theory of Fréchet means in some infinite-dimensional metric spaces, which allows us to recover, strengthen, and generalize most existing results; in particular, we develop novel asymptotic theory for Fréchet means in some infinite-dimensional metric spaces from statistical shape analysis. The core of the proof is a novel notion of weak convergence in general metric spaces for which the results can be proven via calculus of variations.