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 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.
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://archive.numdam.org/articles/10.1051/ps/2013033/} }
TY - JOUR AU - Arnaudon, Marc AU - Miclo, Laurent TI - Means in complete manifolds: uniqueness and approximation JO - ESAIM: Probability and Statistics PY - 2014 SP - 185 EP - 206 VL - 18 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2013033/ DO - 10.1051/ps/2013033 LA - en ID - PS_2014__18__185_0 ER -
%0 Journal Article %A Arnaudon, Marc %A Miclo, Laurent %T Means in complete manifolds: uniqueness and approximation %J ESAIM: Probability and Statistics %D 2014 %P 185-206 %V 18 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2013033/ %R 10.1051/ps/2013033 %G en %F PS_2014__18__185_0
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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