{"paper":{"title":"The Topology of Probability Distributions on Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Omer Bobrowski, Sayan Mukherjee","submitted_at":"2013-07-03T19:57:54Z","abstract_excerpt":"Let $P$ be a set of $n$ random points in $R^d$, generated from a probability measure on a $m$-dimensional manifold $M \\subset R^d$. In this paper we study the homology of $U(P,r)$ -- the union of $d$-dimensional balls of radius $r$ around $P$, as $n \\to \\infty$, and $r \\to 0$. In addition we study the critical points of $d_P$ -- the distance function from the set $P$. These two objects are known to be related via Morse theory. We present limit theorems for the Betti numbers of $U(P,r)$, as well as for number of critical points of index $k$ for $d_P$. Depending on how fast $r$ decays to zero as"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1123","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"}