{"paper":{"title":"The local quantization behavior of absolutely continuous probabilities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Gilles Pag\\`es, Harald Luschgy, Siegfried Graf","submitted_at":"2010-09-26T14:04:54Z","abstract_excerpt":"For a large class of absolutely continuous probabilities $P$ it is shown that, for $r>0$, for $n$-optimal $L^r(P)$-codebooks $\\alpha_n$, and any Voronoi partition $V_{n,a}$ with respect to $\\alpha_n$ the local probabilities $P(V_{n,a})$ satisfy $P(V_{a,n})\\approx n^{-1}$ while the local $L^r$-quantization errors satisfy $\\int_{V_{n,a}}|x-a|^r dP(x)\\approx n^{-(1+r/d)}$ as long as the partition sets $V_{n,a}$ intersect a fixed compact set $K$ in the interior of the support of $P$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.5093","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"}