Necessary and sufficient condition for the existence of a Fréchet mean on the circle

Abstract

Let ( \hbox{}S1,dS1 ) be the unit circle in ℝ2 endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure μ to admit a well defined Frechet mean on ( \hbox{}S1,dS1 ). We derive a new sufficient condition of existence P(α, ϕ) with no restriction on the support of the measure. Then, we study the convergence of the empirical Frechet mean to the Frechet mean and we give an algorithm to compute it.