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

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.