{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:WBSAXYKJ7X2UAGMUKM5BHFBX4M","short_pith_number":"pith:WBSAXYKJ","schema_version":"1.0","canonical_sha256":"b0640be149fdf5401994533a139437e33642457c8b002bab23bad759a82bcb52","source":{"kind":"arxiv","id":"1807.08365","version":2},"attestation_state":"computed","paper":{"title":"On the rate of convergence of empirical measure in $\\infty-$Wasserstein distance for unbounded density function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Anning Liu, Jian-Guo Liu, Yulong Lu","submitted_at":"2018-07-22T20:57:17Z","abstract_excerpt":"We consider a sequence of identically independently distributed random samples from an absolutely continuous probability measure in one dimension with unbounded density. We establish a new rate of convergence of the $\\infty-$Wasserstein distance between the empirical measure of the samples and the true distribution, which extends the previous convergence result by Trilllos and Slep\\v{c}ev to the case that the true distribution has an unbounded density."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1807.08365","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-07-22T20:57:17Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"fbe2a31f731813a9390221523c2ac4a0c090a28545eb4b7e6aecaae3826a2f14","abstract_canon_sha256":"0299a821eb724a509f54e5fb6cb142901ae847b67bfb31a71152426aed1ff4c3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:05.335207Z","signature_b64":"nTKkRtmKS9Z84b5Q6Zu5kI2BM48NQFuyi8kpEq+6vdmncvvy8CHP6x6n5bnKK/i7x5eA5A/y79qmZ6AqajnoBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0640be149fdf5401994533a139437e33642457c8b002bab23bad759a82bcb52","last_reissued_at":"2026-05-18T00:09:05.334530Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:05.334530Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the rate of convergence of empirical measure in $\\infty-$Wasserstein distance for unbounded density function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Anning Liu, Jian-Guo Liu, Yulong Lu","submitted_at":"2018-07-22T20:57:17Z","abstract_excerpt":"We consider a sequence of identically independently distributed random samples from an absolutely continuous probability measure in one dimension with unbounded density. We establish a new rate of convergence of the $\\infty-$Wasserstein distance between the empirical measure of the samples and the true distribution, which extends the previous convergence result by Trilllos and Slep\\v{c}ev to the case that the true distribution has an unbounded density."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.08365","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1807.08365","created_at":"2026-05-18T00:09:05.334622+00:00"},{"alias_kind":"arxiv_version","alias_value":"1807.08365v2","created_at":"2026-05-18T00:09:05.334622+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.08365","created_at":"2026-05-18T00:09:05.334622+00:00"},{"alias_kind":"pith_short_12","alias_value":"WBSAXYKJ7X2U","created_at":"2026-05-18T12:32:59.047623+00:00"},{"alias_kind":"pith_short_16","alias_value":"WBSAXYKJ7X2UAGMU","created_at":"2026-05-18T12:32:59.047623+00:00"},{"alias_kind":"pith_short_8","alias_value":"WBSAXYKJ","created_at":"2026-05-18T12:32:59.047623+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2502.02623","citing_title":"Sample Complexity of Bias Detection with Subsampled Point-to-Subspace Distances","ref_index":10,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WBSAXYKJ7X2UAGMUKM5BHFBX4M","json":"https://pith.science/pith/WBSAXYKJ7X2UAGMUKM5BHFBX4M.json","graph_json":"https://pith.science/api/pith-number/WBSAXYKJ7X2UAGMUKM5BHFBX4M/graph.json","events_json":"https://pith.science/api/pith-number/WBSAXYKJ7X2UAGMUKM5BHFBX4M/events.json","paper":"https://pith.science/paper/WBSAXYKJ"},"agent_actions":{"view_html":"https://pith.science/pith/WBSAXYKJ7X2UAGMUKM5BHFBX4M","download_json":"https://pith.science/pith/WBSAXYKJ7X2UAGMUKM5BHFBX4M.json","view_paper":"https://pith.science/paper/WBSAXYKJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1807.08365&json=true","fetch_graph":"https://pith.science/api/pith-number/WBSAXYKJ7X2UAGMUKM5BHFBX4M/graph.json","fetch_events":"https://pith.science/api/pith-number/WBSAXYKJ7X2UAGMUKM5BHFBX4M/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WBSAXYKJ7X2UAGMUKM5BHFBX4M/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WBSAXYKJ7X2UAGMUKM5BHFBX4M/action/storage_attestation","attest_author":"https://pith.science/pith/WBSAXYKJ7X2UAGMUKM5BHFBX4M/action/author_attestation","sign_citation":"https://pith.science/pith/WBSAXYKJ7X2UAGMUKM5BHFBX4M/action/citation_signature","submit_replication":"https://pith.science/pith/WBSAXYKJ7X2UAGMUKM5BHFBX4M/action/replication_record"}},"created_at":"2026-05-18T00:09:05.334622+00:00","updated_at":"2026-05-18T00:09:05.334622+00:00"}