Stochastic algorithms for computing means of probability measures

Consider a probability measure μ supported by a regular geodesic ball in a manifold. For any p ≥ 1 we define a stochastic algorithm which converges almost surely to the p -mean e p of μ . Assuming furthermore that the functional to minimize is regular around e p , we prove that a natural renormalization of the inhomogeneous Markov chain converges in law into an inhomogeneous diffusion process. We give an explicit expression of this process, as well as its local characteristic.