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Article Dans Une Revue ESAIM: Probability and Statistics Année : 2014

Means in complete manifolds: uniqueness and approximation

Résumé

Let $M$ be a complete Riemannian manifold, $N\in \NN$ and $p\ge 1$. We prove that almost everywhere on $x=(x_1,\ldots,x_N)\in M^N$ for Lebesgue measure in $M^N$, the measure $\di \mu(x)=\f1N\sum_{k=1}^N\d_{x_k}$ has a unique $p$-mean $e_p(x)$. As a consequence, if $X=(X_1,\ldots,X_N)$ is a $M^N$-valued random variable with absolutely continuous law, then almost surely $\mu(X(\om))$ has a unique $p$-mean. In particular if $(X_n)_{n\ge 1}$ is an independent sample of an absolutely continuous law in $M$, then the process $e_{p,n}(\om)=e_p(X_1(\om),\ldots, X_n(\om))$ is well-defined. Assume $M$ is compact and consider a probability measure $\nu$ in $M$. Using partial simulated annealing, we define a continuous semimartingale which converges to the set of minimizers of the integral of distance at power~$p$ with respect to $\nu$. When the set is a singleton, it converges to the $p$-mean.
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Dates et versions

hal-00717677 , version 1 (13-07-2012)

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Marc Arnaudon, Laurent Miclo. Means in complete manifolds: uniqueness and approximation. ESAIM: Probability and Statistics, 2014, 18, pp.185-206. ⟨hal-00717677⟩
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