Means in complete manifolds: uniqueness and approximation
ESAIM: Probability and Statistics, Tome 18 (2014), pp. 185-206.

Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = (x1,...,xN) ∈ MN for Lebesgue measure in MN, the measure d μ ( x ) = 1 N k = 1 N δ x k has a unique p-mean ep(x). As a consequence, if X = (X1,...,XN) is a MN-valued random variable with absolutely continuous law, then almost surely μ(X(ω)) has a unique p-mean. In particular if (Xn)n ≥ 1 is an independent sample of an absolutely continuous law in M, then the process ep,n(ω) = ep(X1(ω),...,Xn(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.

DOI : 10.1051/ps/2013033
Classification : 60D05, 58C35, 37A30, 53C21, 60J65
Mots clés : stochastic algorithms, diffusion processes, simulated annealing, homogenization, probability measures on compact riemannian manifolds, intrinsic p-means, instantaneous invariant measures, Gibbs measures, spectral gap at small temperature
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     title = {Means in complete manifolds: uniqueness and approximation},
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     url = {http://archive.numdam.org/articles/10.1051/ps/2013033/}
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Arnaudon, Marc; Miclo, Laurent. Means in complete manifolds: uniqueness and approximation. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 185-206. doi : 10.1051/ps/2013033. http://archive.numdam.org/articles/10.1051/ps/2013033/

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