Stochastic algorithms for computing means of probability measures

Consider a probability measure mu 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 mu. 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. (c) 2011 Elsevier B.V. All rights reserved.