{"paper":{"title":"Means in complete manifolds: uniqueness and approximation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Laurent Miclo (IMT), Marc Arnaudon (LMA)","submitted_at":"2012-07-13T13:12:29Z","abstract_excerpt":"Let $M$ be a complete Riemannian manifold, $N\\in \\NN$ and $p\\ge 1$. We prove that almost everywhere on $x=(x_1,...,x_N)\\in M^N$ 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 $X=(X_1,...,X_N)$ 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 $e_{p,n}(\\om)=e_p(X_1(\\om),..., X_n(\\om))$ is well-defined. Assume $M$ is compact "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3232","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}