# The Logarithm Map, its Limits and Frechet Means in Orthant Spaces

## TL;DR

This paper investigates the properties of the logarithm map in orthant spaces and uses these insights to analyze Frechet means, including their characterization and limiting distributions, advancing understanding in stratified metric spaces.

## Contribution

It provides a detailed analysis of the logarithm map in orthant spaces and characterizes Frechet means, including their asymptotic behavior, which was previously not well-understood.

## Key findings

- Derived explicit expressions for the logarithm map in orthant spaces.
- Characterized the Frechet means in stratified spaces.
- Established the limiting distribution of sample Frechet means.

## Abstract

The first part of the paper studies the expression for, and the properties of, the logarithm map on an orthant space, which is a simple stratified space, with the aim of analysing Frechet means of probability measures on such a space. In the second part, we use these results to characterise Frechet means and to derive various of their properties, including the limiting distribution of sample Frechet means.