Means in complete manifolds: uniqueness and approximation. (27th March 2014) - Arnaudon, Marc; Miclo, Laurent
Abstract : Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = ( x 1, ..., x N ) ∈ M N for Lebesgue measure in M N, the measureμ(x)=1N_k=1^N_x_k μ ( x ) = 1 N ∑ k = 1 N δ x k has a unique p –mean e p ( x ). As a consequence, if X = ( X 1, ..., X N ) is a M N -valued random variable with absolutely continuous law, then almost surely μ ( X ( ω )) has a unique p –mean. In particular if ( X n ) n ≥ 1 is an independent sample of an absolutely continuous law in M, then the process e p, n ( ω ) = e p ( X 1 ( ω ), ..., X n ( ω )) is well-defined. Assume M is compact and consider a probability measure ν in M . Using partial simulated annealing, we define a continuous semimartingale which converges in probability to the set of minimizers of the integral of distance at power p with respect to ν . When the set is a singleton, it converges to the p –mean.