{"paper":{"title":"Limit theory for point processes in manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"J. E. Yukich, Mathew D. Penrose","submitted_at":"2011-04-05T19:04:24Z","abstract_excerpt":"Let $Y_i,i\\geq1$, be i.i.d. random variables having values in an $m$-dimensional manifold $\\mathcal {M}\\subset \\mathbb{R}^d$ and consider sums $\\sum_{i=1}^n\\xi(n^{1/m}Y_i,\\{n^{1/m}Y_j\\}_{j=1}^n)$, where $\\xi$ is a real valued function defined on pairs $(y,\\mathcal {Y})$, with $y\\in \\mathbb{R}^d$ and $\\mathcal {Y}\\subset \\mathbb{R}^d$ locally finite. Subject to $\\xi$ satisfying a weak spatial dependence and continuity condition, we show that such sums satisfy weak laws of large numbers, variance asymptotics and central limit theorems. We show that the limit behavior is controlled by the value o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.0914","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"}