February 2024 Limit theorems for Fréchet mean sets
Steven N. Evans, Adam Q. Jaffe
Author Affiliations +
Bernoulli 30(1): 419-447 (February 2024). DOI: 10.3150/23-BEJ1603

Abstract

For 1p, the Fréchet p-mean of a probability measure on a metric space is an important notion of central tendency that generalizes the usual notions in the real line of mean (p=2) and median (p=1). In this work we prove a collection of limit theorems for Fréchet means and related objects, which, in general, constitute a sequence of random closed sets. On the one hand, we show that many limit theorems (a strong law of large numbers, an ergodic theorem, and a large deviations principle) can be simply descended from analogous theorems on the space of probability measures via purely topological considerations. On the other hand, we provide the first sufficient conditions for the strong law of large numbers to hold in a T2 topology (in particular, the Fell topology), and we show that this condition is necessary in some special cases. We also discuss statistical and computational implications of the results herein.

Citation

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Steven N. Evans. Adam Q. Jaffe. "Limit theorems for Fréchet mean sets." Bernoulli 30 (1) 419 - 447, February 2024. https://doi.org/10.3150/23-BEJ1603

Information

Received: 1 December 2020; Published: February 2024
First available in Project Euclid: 8 November 2023

MathSciNet: MR4665584
Digital Object Identifier: 10.3150/23-BEJ1603

Keywords: Hausdorff metric , Karcher mean , Kuratowski convergence , medoids , non-Euclidean statistics , random sets , Wasserstein metric

Vol.30 • No. 1 • February 2024
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