{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:3OYXTNUO55D5UTVHYUCOEDJ5HU","short_pith_number":"pith:3OYXTNUO","canonical_record":{"source":{"id":"1301.0726","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2013-01-04T13:58:12Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"c66c9a7c69ca1428666e5a5d039e1b6da1c4a3d475c57e5c002cf95d55f7872c","abstract_canon_sha256":"a7ebd1abf4e8e0aab0f8e0cd03359266e67704764d213342617049a34afe635b"},"schema_version":"1.0"},"canonical_sha256":"dbb179b68eef47da4ea7c504e20d3d3d398458e278ebf3f3311434648361e47f","source":{"kind":"arxiv","id":"1301.0726","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.0726","created_at":"2026-05-18T03:37:17Z"},{"alias_kind":"arxiv_version","alias_value":"1301.0726v1","created_at":"2026-05-18T03:37:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.0726","created_at":"2026-05-18T03:37:17Z"},{"alias_kind":"pith_short_12","alias_value":"3OYXTNUO55D5","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"3OYXTNUO55D5UTVH","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"3OYXTNUO","created_at":"2026-05-18T12:27:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:3OYXTNUO55D5UTVHYUCOEDJ5HU","target":"record","payload":{"canonical_record":{"source":{"id":"1301.0726","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2013-01-04T13:58:12Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"c66c9a7c69ca1428666e5a5d039e1b6da1c4a3d475c57e5c002cf95d55f7872c","abstract_canon_sha256":"a7ebd1abf4e8e0aab0f8e0cd03359266e67704764d213342617049a34afe635b"},"schema_version":"1.0"},"canonical_sha256":"dbb179b68eef47da4ea7c504e20d3d3d398458e278ebf3f3311434648361e47f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:37:17.821973Z","signature_b64":"xeQkSIHu8wjnDIOtlyVClfN0ERDfggPDLGRfX8MZSWxleTcaHy8XDdSWXLPs/EAYSJGhP0A3K6cJz2ClgTbTAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dbb179b68eef47da4ea7c504e20d3d3d398458e278ebf3f3311434648361e47f","last_reissued_at":"2026-05-18T03:37:17.821273Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:37:17.821273Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1301.0726","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":"2026-05-18T03:37:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Cd165VCZgS81gvsPvt41FK0X4vvAa+8MbrPdO8BHxV5XIJh/+fYjaBhwqDwMt0N3qIG/8AFKGTXy63kVDGtMCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T15:47:45.142918Z"},"content_sha256":"7b81bbabb11382b11f796c870066c23606d34090b3e056bf944a4cd14d5a9551","schema_version":"1.0","event_id":"sha256:7b81bbabb11382b11f796c870066c23606d34090b3e056bf944a4cd14d5a9551"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:3OYXTNUO55D5UTVHYUCOEDJ5HU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Marcinkiewicz-Zygmund and ordinary strong laws for empirical distribution functions and plug-in estimators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Henryk Z\\\"ahle","submitted_at":"2013-01-04T13:58:12Z","abstract_excerpt":"Both Marcinkiewicz-Zygmund strong laws of large numbers (MZ-SLLNs) and ordinary strong laws of large numbers (SLLNs) for plug-in estimators of general statistical functionals are derived. It is used that if a statistical functional is \"sufficiently regular\", then a (MZ-) SLLN for the estimator of the unknown distribution function yields a (MZ-) SLLN for the corresponding plug-in estimator. It is in particular shown that many L-, V- and risk functionals are \"sufficiently regular\", and that known results on the strong convergence of the empirical process of \\alpha-mixing random variables can be "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.0726","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":"2026-05-18T03:37:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8KYkMhLEXFg5xnvSYkn85fAUQy0JbehfIL9Zek6YEBYAyG8RAYa5V0omLu/6ETwBL2RJARnprlaXjj0436iTBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T15:47:45.143259Z"},"content_sha256":"761e8a65efb014142266e0364f0312ccef303ccf532317295545d0004f171662","schema_version":"1.0","event_id":"sha256:761e8a65efb014142266e0364f0312ccef303ccf532317295545d0004f171662"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3OYXTNUO55D5UTVHYUCOEDJ5HU/bundle.json","state_url":"https://pith.science/pith/3OYXTNUO55D5UTVHYUCOEDJ5HU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3OYXTNUO55D5UTVHYUCOEDJ5HU/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":"2026-06-22T15:47:45Z","links":{"resolver":"https://pith.science/pith/3OYXTNUO55D5UTVHYUCOEDJ5HU","bundle":"https://pith.science/pith/3OYXTNUO55D5UTVHYUCOEDJ5HU/bundle.json","state":"https://pith.science/pith/3OYXTNUO55D5UTVHYUCOEDJ5HU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3OYXTNUO55D5UTVHYUCOEDJ5HU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:3OYXTNUO55D5UTVHYUCOEDJ5HU","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":"a7ebd1abf4e8e0aab0f8e0cd03359266e67704764d213342617049a34afe635b","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2013-01-04T13:58:12Z","title_canon_sha256":"c66c9a7c69ca1428666e5a5d039e1b6da1c4a3d475c57e5c002cf95d55f7872c"},"schema_version":"1.0","source":{"id":"1301.0726","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.0726","created_at":"2026-05-18T03:37:17Z"},{"alias_kind":"arxiv_version","alias_value":"1301.0726v1","created_at":"2026-05-18T03:37:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.0726","created_at":"2026-05-18T03:37:17Z"},{"alias_kind":"pith_short_12","alias_value":"3OYXTNUO55D5","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"3OYXTNUO55D5UTVH","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"3OYXTNUO","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:761e8a65efb014142266e0364f0312ccef303ccf532317295545d0004f171662","target":"graph","created_at":"2026-05-18T03:37:17Z","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":"Both Marcinkiewicz-Zygmund strong laws of large numbers (MZ-SLLNs) and ordinary strong laws of large numbers (SLLNs) for plug-in estimators of general statistical functionals are derived. It is used that if a statistical functional is \"sufficiently regular\", then a (MZ-) SLLN for the estimator of the unknown distribution function yields a (MZ-) SLLN for the corresponding plug-in estimator. It is in particular shown that many L-, V- and risk functionals are \"sufficiently regular\", and that known results on the strong convergence of the empirical process of \\alpha-mixing random variables can be ","authors_text":"Henryk Z\\\"ahle","cross_cats":["stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2013-01-04T13:58:12Z","title":"Marcinkiewicz-Zygmund and ordinary strong laws for empirical distribution functions and plug-in estimators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.0726","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":"sha256:7b81bbabb11382b11f796c870066c23606d34090b3e056bf944a4cd14d5a9551","target":"record","created_at":"2026-05-18T03:37:17Z","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":"a7ebd1abf4e8e0aab0f8e0cd03359266e67704764d213342617049a34afe635b","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2013-01-04T13:58:12Z","title_canon_sha256":"c66c9a7c69ca1428666e5a5d039e1b6da1c4a3d475c57e5c002cf95d55f7872c"},"schema_version":"1.0","source":{"id":"1301.0726","kind":"arxiv","version":1}},"canonical_sha256":"dbb179b68eef47da4ea7c504e20d3d3d398458e278ebf3f3311434648361e47f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dbb179b68eef47da4ea7c504e20d3d3d398458e278ebf3f3311434648361e47f","first_computed_at":"2026-05-18T03:37:17.821273Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:37:17.821273Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xeQkSIHu8wjnDIOtlyVClfN0ERDfggPDLGRfX8MZSWxleTcaHy8XDdSWXLPs/EAYSJGhP0A3K6cJz2ClgTbTAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:37:17.821973Z","signed_message":"canonical_sha256_bytes"},"source_id":"1301.0726","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7b81bbabb11382b11f796c870066c23606d34090b3e056bf944a4cd14d5a9551","sha256:761e8a65efb014142266e0364f0312ccef303ccf532317295545d0004f171662"],"state_sha256":"add3049d37fe801cf8c6c512200a7df82f39ec3e10eca590f6676d1f585eac8d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"auP+ctmpN64vIVVcUFEAkEgFfNzrgGxOUkPX7fyg70G/T50bew/FidM6z8ITujlOGtZd5z9edPRN5vRxQ50fAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T15:47:45.145195Z","bundle_sha256":"3009a2194d109c3161a88a572f5fdee15485f25fe706393f2cbe67a0fc5739dc"}}