 
Summary: STOCHASTIC ALGORITHMS FOR COMPUTING MEANS OF
PROBABILITY MEASURES
MARC ARNAUDON, CL´EMENT DOMBRY, ANTHONY PHAN, AND LE YANG
Abstract. Consider a probability measure µ supported by a regular geodesic
ball in a manifold. For any p 1 we define a stochastic algorithm which con
verges almost surely to the pmean ep of µ. Assuming furthermore that the
functional to minimize is regular around ep, we prove that a natural renormal
ization of the inhomogeneous Markov chain converges in law into an inhomo
geneous diffusion process. We give an explicit expression of this process, as
well as its local characteristic.
1. Introduction
Consider a set of points {x1, . . . , xn} in an Euclidean space E with metric d.
The geometric barycenter e2 of this set of points is the unique point minimizing
the mean square distance to these points, i.e.
e2 = arg min
xE
1
n
n
i=1
