{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:OHVUAOUI2GAUEXDJSK2QJMR6JP","short_pith_number":"pith:OHVUAOUI","canonical_record":{"source":{"id":"1102.0228","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-01T17:30:37Z","cross_cats_sorted":[],"title_canon_sha256":"f3a9b356e4e24af4f5ca07cbbdcc2f87c7d473249d514a061cbf224f3380bc08","abstract_canon_sha256":"6ccbe1fa4a514bdcd61555527c4df5a2f885f8d0628d1ac29b784d47cc43df09"},"schema_version":"1.0"},"canonical_sha256":"71eb403a88d181425c6992b504b23e4bc9e4bca4f13787fab7962f22ccd6ea44","source":{"kind":"arxiv","id":"1102.0228","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.0228","created_at":"2026-05-18T04:19:13Z"},{"alias_kind":"arxiv_version","alias_value":"1102.0228v2","created_at":"2026-05-18T04:19:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.0228","created_at":"2026-05-18T04:19:13Z"},{"alias_kind":"pith_short_12","alias_value":"OHVUAOUI2GAU","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"OHVUAOUI2GAUEXDJ","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"OHVUAOUI","created_at":"2026-05-18T12:26:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:OHVUAOUI2GAUEXDJSK2QJMR6JP","target":"record","payload":{"canonical_record":{"source":{"id":"1102.0228","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-01T17:30:37Z","cross_cats_sorted":[],"title_canon_sha256":"f3a9b356e4e24af4f5ca07cbbdcc2f87c7d473249d514a061cbf224f3380bc08","abstract_canon_sha256":"6ccbe1fa4a514bdcd61555527c4df5a2f885f8d0628d1ac29b784d47cc43df09"},"schema_version":"1.0"},"canonical_sha256":"71eb403a88d181425c6992b504b23e4bc9e4bca4f13787fab7962f22ccd6ea44","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:19:13.214222Z","signature_b64":"8HpYAAFncblaNt7Q3gNVnyDWB/f5OshiAWmnDrF/sA3mG34gR2RuOYGsLChzNBWMvhJ83iL6K9r50gp5ZOMSAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71eb403a88d181425c6992b504b23e4bc9e4bca4f13787fab7962f22ccd6ea44","last_reissued_at":"2026-05-18T04:19:13.213805Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:19:13.213805Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1102.0228","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:19:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FM7gxcDNqnKfMMGIIPIs+GaDPi6cEdg4CQiSPEJ/8cMAaFVdJbvvL0Fi1ujUhBQALPQrxoRdSbuIsfn/lmZRAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T18:19:37.381489Z"},"content_sha256":"c40461ec67bfebf2354fc3e33efa4a268d14fef1acc4195cc34126b6ce23583d","schema_version":"1.0","event_id":"sha256:c40461ec67bfebf2354fc3e33efa4a268d14fef1acc4195cc34126b6ce23583d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:OHVUAOUI2GAUEXDJSK2QJMR6JP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Limit theorems for empirical Fr\\'echet means of independent and non-identically distributed manifold-valued random variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Huiling Le, Wilfrid S. Kendall","submitted_at":"2011-02-01T17:30:37Z","abstract_excerpt":"We prove weak laws of large numbers and central limit theorems of Lindeberg type for empirical centres of mass (empirical Fr\\'echet means) of independent non-identically distributed random variables taking values in Riemannian manifolds. In order to prove these theorems we describe and prove a simple kind of Lindeberg-Feller central approximation theorem for vector-valued random variables, which may be of independent interest and is therefore the subject of a self-contained section. This vector-valued result allows us to clarify the number of conditions required for the central limit theorem f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.0228","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:19:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"svSnwTOvarAjda9Dze4k5KM5/hLQg97a4cbI1sXvfH4F/mBNScJvEvGb34pRzU4IxkJHCGaaD1h1Mwkemh5bDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T18:19:37.381835Z"},"content_sha256":"77ca3b942c9b6816e8b7251060ac289c36da92367afdb02019f19366fd482604","schema_version":"1.0","event_id":"sha256:77ca3b942c9b6816e8b7251060ac289c36da92367afdb02019f19366fd482604"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OHVUAOUI2GAUEXDJSK2QJMR6JP/bundle.json","state_url":"https://pith.science/pith/OHVUAOUI2GAUEXDJSK2QJMR6JP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OHVUAOUI2GAUEXDJSK2QJMR6JP/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":"2026-06-23T18:19:37Z","links":{"resolver":"https://pith.science/pith/OHVUAOUI2GAUEXDJSK2QJMR6JP","bundle":"https://pith.science/pith/OHVUAOUI2GAUEXDJSK2QJMR6JP/bundle.json","state":"https://pith.science/pith/OHVUAOUI2GAUEXDJSK2QJMR6JP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OHVUAOUI2GAUEXDJSK2QJMR6JP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:OHVUAOUI2GAUEXDJSK2QJMR6JP","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":"6ccbe1fa4a514bdcd61555527c4df5a2f885f8d0628d1ac29b784d47cc43df09","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-01T17:30:37Z","title_canon_sha256":"f3a9b356e4e24af4f5ca07cbbdcc2f87c7d473249d514a061cbf224f3380bc08"},"schema_version":"1.0","source":{"id":"1102.0228","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.0228","created_at":"2026-05-18T04:19:13Z"},{"alias_kind":"arxiv_version","alias_value":"1102.0228v2","created_at":"2026-05-18T04:19:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.0228","created_at":"2026-05-18T04:19:13Z"},{"alias_kind":"pith_short_12","alias_value":"OHVUAOUI2GAU","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"OHVUAOUI2GAUEXDJ","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"OHVUAOUI","created_at":"2026-05-18T12:26:37Z"}],"graph_snapshots":[{"event_id":"sha256:77ca3b942c9b6816e8b7251060ac289c36da92367afdb02019f19366fd482604","target":"graph","created_at":"2026-05-18T04:19:13Z","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":"We prove weak laws of large numbers and central limit theorems of Lindeberg type for empirical centres of mass (empirical Fr\\'echet means) of independent non-identically distributed random variables taking values in Riemannian manifolds. In order to prove these theorems we describe and prove a simple kind of Lindeberg-Feller central approximation theorem for vector-valued random variables, which may be of independent interest and is therefore the subject of a self-contained section. This vector-valued result allows us to clarify the number of conditions required for the central limit theorem f","authors_text":"Huiling Le, Wilfrid S. Kendall","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-01T17:30:37Z","title":"Limit theorems for empirical Fr\\'echet means of independent and non-identically distributed manifold-valued random variables"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.0228","kind":"arxiv","version":2},"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:c40461ec67bfebf2354fc3e33efa4a268d14fef1acc4195cc34126b6ce23583d","target":"record","created_at":"2026-05-18T04:19:13Z","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":"6ccbe1fa4a514bdcd61555527c4df5a2f885f8d0628d1ac29b784d47cc43df09","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-01T17:30:37Z","title_canon_sha256":"f3a9b356e4e24af4f5ca07cbbdcc2f87c7d473249d514a061cbf224f3380bc08"},"schema_version":"1.0","source":{"id":"1102.0228","kind":"arxiv","version":2}},"canonical_sha256":"71eb403a88d181425c6992b504b23e4bc9e4bca4f13787fab7962f22ccd6ea44","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"71eb403a88d181425c6992b504b23e4bc9e4bca4f13787fab7962f22ccd6ea44","first_computed_at":"2026-05-18T04:19:13.213805Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:19:13.213805Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8HpYAAFncblaNt7Q3gNVnyDWB/f5OshiAWmnDrF/sA3mG34gR2RuOYGsLChzNBWMvhJ83iL6K9r50gp5ZOMSAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:19:13.214222Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.0228","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c40461ec67bfebf2354fc3e33efa4a268d14fef1acc4195cc34126b6ce23583d","sha256:77ca3b942c9b6816e8b7251060ac289c36da92367afdb02019f19366fd482604"],"state_sha256":"e973db4231e9a61f574af482d2bc15f965841e741ffee803d35fc134a0e58bfb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VV+lsvM5eNr+gn5JpprRiVfP/6Q+1h96qGG6JgZpfTjVrWrLL/6E51KaJYIa+Jb6pcoRVYSU6J8hGYXiiZzTDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T18:19:37.383805Z","bundle_sha256":"aa933f0ea347b14733a67c557274561abfe31244239cafe4dc3bf8faf8104da2"}}