{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2004:KNNNOZFKHQVUE5ESV6FZ6SPFXM","short_pith_number":"pith:KNNNOZFK","canonical_record":{"source":{"id":"math/0410170","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2004-10-06T15:52:22Z","cross_cats_sorted":[],"title_canon_sha256":"18ee338b91c41262add71d6974fde90a3cebb8381cd2b63d8559e2c50cc88e1d","abstract_canon_sha256":"09dc4b0ff7fcc9d040c3a5dc6714d6c3f384a41cdb227677f5f6d76596dbcf44"},"schema_version":"1.0"},"canonical_sha256":"535ad764aa3c2b427492af8b9f49e5bb256001b402f555cf9957f4bc7e6286f9","source":{"kind":"arxiv","id":"math/0410170","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0410170","created_at":"2026-05-18T01:05:25Z"},{"alias_kind":"arxiv_version","alias_value":"math/0410170v1","created_at":"2026-05-18T01:05:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0410170","created_at":"2026-05-18T01:05:25Z"},{"alias_kind":"pith_short_12","alias_value":"KNNNOZFKHQVU","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"KNNNOZFKHQVUE5ES","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"KNNNOZFK","created_at":"2026-05-18T12:25:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2004:KNNNOZFKHQVUE5ESV6FZ6SPFXM","target":"record","payload":{"canonical_record":{"source":{"id":"math/0410170","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2004-10-06T15:52:22Z","cross_cats_sorted":[],"title_canon_sha256":"18ee338b91c41262add71d6974fde90a3cebb8381cd2b63d8559e2c50cc88e1d","abstract_canon_sha256":"09dc4b0ff7fcc9d040c3a5dc6714d6c3f384a41cdb227677f5f6d76596dbcf44"},"schema_version":"1.0"},"canonical_sha256":"535ad764aa3c2b427492af8b9f49e5bb256001b402f555cf9957f4bc7e6286f9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:25.764481Z","signature_b64":"KUFsU7TJIdzSZ236aEm5Fkhh9OlUh1XOXnPRfPUA3f1lAoRvwmYVRjToZ4hd5jpfuXRY4/mgR8dryHbppx2KCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"535ad764aa3c2b427492af8b9f49e5bb256001b402f555cf9957f4bc7e6286f9","last_reissued_at":"2026-05-18T01:05:25.763991Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:25.763991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0410170","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:05:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/9SngnwPbt/ecarCeVEG3lavQWXmSLX6jXq6z/9oVot8dp3JDT58q77AU2u1yxyCgkFaEVUkEcBZct9Tiku5DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T18:22:36.690137Z"},"content_sha256":"66fe508b7aedfcd0bcdc16bf561767b7daa1890595879c67aaae19ba32150100","schema_version":"1.0","event_id":"sha256:66fe508b7aedfcd0bcdc16bf561767b7daa1890595879c67aaae19ba32150100"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2004:KNNNOZFKHQVUE5ESV6FZ6SPFXM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weighted uniform consistency of kernel density estimators","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Evarist Gine, Joel Zinn, Vladimir Koltchinskii","submitted_at":"2004-10-06T15:52:22Z","abstract_excerpt":"Let f_n denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let \\Psi(t) be a positive continuous function such that \\|\\Psi f^{\\beta}\\|_{\\infty}<\\infty for some 0<\\beta<1/2.\n Under natural smoothness conditions, necessary and sufficient conditions for the sequence \\sqrt\\frac{nh_n^d}{2|\\log h_n^d|}\\|\\Psi(t)(f_n(t)-Ef_n(t))\\|_{\\infty} to be stochastically bounded and to converge a.s. to a constant are obtained.\n Also, the case of larger values of \\beta is studied where a similar sequence with a different norming converges a.s. either to 0 or to +\\inf"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0410170","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:05:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iwE2lRPwgAsc8jZ1ZDkScZF7wtMdzCIG5Csfe4VA2SHQTB1DPZxMxwngEmItHNacYik+yEpsmeiWOu+nyZvpDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T18:22:36.690843Z"},"content_sha256":"0336f7aef669c1281fa9031dc9955a6f13274b98b48be796dd7c22e221255ecb","schema_version":"1.0","event_id":"sha256:0336f7aef669c1281fa9031dc9955a6f13274b98b48be796dd7c22e221255ecb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KNNNOZFKHQVUE5ESV6FZ6SPFXM/bundle.json","state_url":"https://pith.science/pith/KNNNOZFKHQVUE5ESV6FZ6SPFXM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KNNNOZFKHQVUE5ESV6FZ6SPFXM/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-31T18:22:36Z","links":{"resolver":"https://pith.science/pith/KNNNOZFKHQVUE5ESV6FZ6SPFXM","bundle":"https://pith.science/pith/KNNNOZFKHQVUE5ESV6FZ6SPFXM/bundle.json","state":"https://pith.science/pith/KNNNOZFKHQVUE5ESV6FZ6SPFXM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KNNNOZFKHQVUE5ESV6FZ6SPFXM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:KNNNOZFKHQVUE5ESV6FZ6SPFXM","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"09dc4b0ff7fcc9d040c3a5dc6714d6c3f384a41cdb227677f5f6d76596dbcf44","cross_cats_sorted":[],"license":"","primary_cat":"math.PR","submitted_at":"2004-10-06T15:52:22Z","title_canon_sha256":"18ee338b91c41262add71d6974fde90a3cebb8381cd2b63d8559e2c50cc88e1d"},"schema_version":"1.0","source":{"id":"math/0410170","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0410170","created_at":"2026-05-18T01:05:25Z"},{"alias_kind":"arxiv_version","alias_value":"math/0410170v1","created_at":"2026-05-18T01:05:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0410170","created_at":"2026-05-18T01:05:25Z"},{"alias_kind":"pith_short_12","alias_value":"KNNNOZFKHQVU","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"KNNNOZFKHQVUE5ES","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"KNNNOZFK","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:0336f7aef669c1281fa9031dc9955a6f13274b98b48be796dd7c22e221255ecb","target":"graph","created_at":"2026-05-18T01:05:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let f_n denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let \\Psi(t) be a positive continuous function such that \\|\\Psi f^{\\beta}\\|_{\\infty}<\\infty for some 0<\\beta<1/2.\n Under natural smoothness conditions, necessary and sufficient conditions for the sequence \\sqrt\\frac{nh_n^d}{2|\\log h_n^d|}\\|\\Psi(t)(f_n(t)-Ef_n(t))\\|_{\\infty} to be stochastically bounded and to converge a.s. to a constant are obtained.\n Also, the case of larger values of \\beta is studied where a similar sequence with a different norming converges a.s. either to 0 or to +\\inf","authors_text":"Evarist Gine, Joel Zinn, Vladimir Koltchinskii","cross_cats":[],"headline":"","license":"","primary_cat":"math.PR","submitted_at":"2004-10-06T15:52:22Z","title":"Weighted uniform consistency of kernel density estimators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0410170","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:66fe508b7aedfcd0bcdc16bf561767b7daa1890595879c67aaae19ba32150100","target":"record","created_at":"2026-05-18T01:05:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"09dc4b0ff7fcc9d040c3a5dc6714d6c3f384a41cdb227677f5f6d76596dbcf44","cross_cats_sorted":[],"license":"","primary_cat":"math.PR","submitted_at":"2004-10-06T15:52:22Z","title_canon_sha256":"18ee338b91c41262add71d6974fde90a3cebb8381cd2b63d8559e2c50cc88e1d"},"schema_version":"1.0","source":{"id":"math/0410170","kind":"arxiv","version":1}},"canonical_sha256":"535ad764aa3c2b427492af8b9f49e5bb256001b402f555cf9957f4bc7e6286f9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"535ad764aa3c2b427492af8b9f49e5bb256001b402f555cf9957f4bc7e6286f9","first_computed_at":"2026-05-18T01:05:25.763991Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:25.763991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KUFsU7TJIdzSZ236aEm5Fkhh9OlUh1XOXnPRfPUA3f1lAoRvwmYVRjToZ4hd5jpfuXRY4/mgR8dryHbppx2KCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:25.764481Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0410170","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:66fe508b7aedfcd0bcdc16bf561767b7daa1890595879c67aaae19ba32150100","sha256:0336f7aef669c1281fa9031dc9955a6f13274b98b48be796dd7c22e221255ecb"],"state_sha256":"a20c33d13c51e9582536867e3a18408c02f93bd8335fa9a86260140a7498a2ef"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k/RXqTe6NgQWfngSrwVTLIzba2Eq4YvmCwQhyltc/GMuu08fxNCLrFhk2BZwO72Lo+4lAks+6iR3MOKHQIQZDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T18:22:36.694713Z","bundle_sha256":"443ecfd3dfb17e08aa542aaa2c7a3428bc18ab6661437286482b1e7d212bdcbc"}}