{"paper":{"title":"Nearest-Neighbor Radii under Dependent Sampling","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Nearest-neighbor radii converge almost surely to zero under polynomial strong mixing, with moment bounds controlled by local intrinsic dimension rather than ambient dimension.","cross_cats":["math.ST","stat.ML","stat.TH"],"primary_cat":"cs.LG","authors_text":"Yilong Hou, Yuanyuan Gao, Zhexiao Lin","submitted_at":"2026-05-14T04:07:05Z","abstract_excerpt":"Nearest-neighbor methods are fundamental to classical and modern machine learning, yet their geometric properties are typically analyzed under independent sampling. In this paper, we study the nearest-neighbor radii under dependent sampling. We consider strong mixing dependent observations and ask whether dependence changes the scale of nearest-neighbor neighborhoods. We establish distribution-free almost sure convergence under polynomial mixing and sharp non-asymptotic moment bounds under geometric mixing. The moment bounds depend on the local intrinsic dimension rather than the ambient dimen"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish distribution-free almost sure convergence under polynomial mixing and sharp non-asymptotic moment bounds under geometric mixing. The moment bounds depend on the local intrinsic dimension rather than the ambient dimension.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The observations satisfy strong mixing conditions (polynomial or geometric decay of dependence) whose rate is known or estimable; if mixing is slower than assumed, the convergence and moment bounds may fail.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Nearest-neighbor radii converge almost surely and obey local-dimension moment bounds under polynomial and geometric mixing dependence.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Nearest-neighbor radii converge almost surely to zero under polynomial strong mixing, with moment bounds controlled by local intrinsic dimension rather than ambient dimension.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"2a9874b532fb78412d069dad2cf196c5369df628d0de334c7a159e5211dd2b29"},"source":{"id":"2605.14343","kind":"arxiv","version":1},"verdict":{"id":"c567701a-e97f-4aa0-9c7e-d275f377e504","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:36:21.409022Z","strongest_claim":"We establish distribution-free almost sure convergence under polynomial mixing and sharp non-asymptotic moment bounds under geometric mixing. The moment bounds depend on the local intrinsic dimension rather than the ambient dimension.","one_line_summary":"Nearest-neighbor radii converge almost surely and obey local-dimension moment bounds under polynomial and geometric mixing dependence.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The observations satisfy strong mixing conditions (polynomial or geometric decay of dependence) whose rate is known or estimable; if mixing is slower than assumed, the convergence and moment bounds may fail.","pith_extraction_headline":"Nearest-neighbor radii converge almost surely to zero under polynomial strong mixing, with moment bounds controlled by local intrinsic dimension rather than ambient dimension."},"references":{"count":59,"sample":[{"doi":"","year":null,"title":"Convergence of distributions generated by stationary stochastic processes , volume =","work_id":"575606fe-eac4-4235-b9ef-c3539ee64510","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"The Functional Law of the Iterated Logarithm for Stationary Strongly Mixing Sequences , volume =","work_id":"67a1d685-55ed-49e4-b270-dbb37542eebd","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"A distribution-free theory of nonparametric regression , year =","work_id":"1a4ef15e-6f8b-419d-936f-246ab56795d5","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Non-asymptotic uniform rates of consistency for k-nn regression , volume =","work_id":"a0e0cd46-edee-4231-9bbf-879d332cabae","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Strong convergence of sums of -mixing random variables with applications to density estimation , volume =","work_id":"51cb3ef1-9f9a-49b7-9b1d-32efc4e34c0d","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":59,"snapshot_sha256":"4de66880fd457e9d88b2a5b325633a2269f3189fe7684a87c3f3f580c6fc72e6","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"7b545d79c97a312e05370fe4fc7e1946425e2cc14cef35fba913cac4cf3d28be"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}