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 $d\mu \left(x\right)=\frac{1}{N}{\sum }_{k=1}^{N}{\delta }_{{x}_{k}}$ 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 : https://doi.org/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
@article{PS_2014__18__185_0,
author = {Arnaudon, Marc and Miclo, Laurent},
title = {Means in complete manifolds: uniqueness and approximation},
journal = {ESAIM: Probability and Statistics},
pages = {185--206},
publisher = {EDP-Sciences},
volume = {18},
year = {2014},
doi = {10.1051/ps/2013033},
mrnumber = {3230874},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2013033/}
}
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://www.numdam.org/articles/10.1051/ps/2013033/

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