{"paper":{"title":"Asymptotics of the maximal radius of an $L^r$-optimal sequence of quantizers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Abass Sagna (LPMA), Gilles Pag\\`es (LPMA)","submitted_at":"2008-06-05T07:40:13Z","abstract_excerpt":"Let $P$ be a probability distribution on $\\mathbb{R}^d$ (equipped with an Euclidean norm $|\\cdot|$). Let $ r> 0 $ and let $(\\alpha_n)_{n \\geq1}$ be an (asymptotically) $L^r(P)$-optimal sequence of $n$-quantizers. We investigate the asymptotic behavior of the maximal radius sequence induced by the sequence $(\\alpha_n)_{n \\geq1}$ defined for every $n \\geq1$ by $\\rho(\\alpha_n) = \\max{|a|, a \\in\\alpha_n}$. When $\\card(\\supp(P))$ is infinite, the maximal radius sequence goes to $\\sup{|x|, x \\in\\operatorname{supp}(P)}$ as $n$ goes to infinity. We then give the exact rate of convergence for two class"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0806.0918","kind":"arxiv","version":2},"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"}