{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:2Z42RRBQT7YEEWBH6HQWM3OZID","short_pith_number":"pith:2Z42RRBQ","canonical_record":{"source":{"id":"1511.05631","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-11-18T00:52:20Z","cross_cats_sorted":[],"title_canon_sha256":"729b07b3d559a77f71fb6c902d2c6f2eec58456bfb0491025886a419493a7d08","abstract_canon_sha256":"08dfa7737d3be8d378d71010b902204977d895fcc69af5d26ba4f302bb755bd3"},"schema_version":"1.0"},"canonical_sha256":"d679a8c4309ff0425827f1e1666dd940d6ffd9d2a2949d3805d64c620320423a","source":{"kind":"arxiv","id":"1511.05631","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.05631","created_at":"2026-07-05T07:13:17Z"},{"alias_kind":"arxiv_version","alias_value":"1511.05631v2","created_at":"2026-07-05T07:13:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.05631","created_at":"2026-07-05T07:13:17Z"},{"alias_kind":"pith_short_12","alias_value":"2Z42RRBQT7YE","created_at":"2026-07-05T07:13:17Z"},{"alias_kind":"pith_short_16","alias_value":"2Z42RRBQT7YEEWBH","created_at":"2026-07-05T07:13:17Z"},{"alias_kind":"pith_short_8","alias_value":"2Z42RRBQ","created_at":"2026-07-05T07:13:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:2Z42RRBQT7YEEWBH6HQWM3OZID","target":"record","payload":{"canonical_record":{"source":{"id":"1511.05631","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-11-18T00:52:20Z","cross_cats_sorted":[],"title_canon_sha256":"729b07b3d559a77f71fb6c902d2c6f2eec58456bfb0491025886a419493a7d08","abstract_canon_sha256":"08dfa7737d3be8d378d71010b902204977d895fcc69af5d26ba4f302bb755bd3"},"schema_version":"1.0"},"canonical_sha256":"d679a8c4309ff0425827f1e1666dd940d6ffd9d2a2949d3805d64c620320423a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:13:17.622574Z","signature_b64":"P6aQizlh8KqWy/IV5wk1jH/oVxSVnA6Vqy4/nlgo6OpZurbNmxNMarxP/JYWdlBBiDCzNTfW1kwEOeq/m85tDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d679a8c4309ff0425827f1e1666dd940d6ffd9d2a2949d3805d64c620320423a","last_reissued_at":"2026-07-05T07:13:17.622148Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:13:17.622148Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.05631","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":"2026-07-05T07:13:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f5Jfw3HDJjxp8Xwc1Sd5jS8vj4pERIA4D7ZD3e6K66CXOsTI9mmNK3qtusuqfkpGr21OvQeMaWS8ZrG432pHBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T11:08:20.446019Z"},"content_sha256":"fa58b5ece569194663fe1bb212e082208efcf5f7928cbbd19a1779ba29fb55f2","schema_version":"1.0","event_id":"sha256:fa58b5ece569194663fe1bb212e082208efcf5f7928cbbd19a1779ba29fb55f2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:2Z42RRBQT7YEEWBH6HQWM3OZID","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Two laws of large numbers for sublinear expectations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Chuanfeng Sun, Wenhao Li","submitted_at":"2015-11-18T00:52:20Z","abstract_excerpt":"In this paper, we consider the sublinear expectation on bounded random variables. With the notion of uncorrelatedness for random variables under the sublinear expectation, a weak law of large numbers is obtained. With the notion of independence for random variable sequences and regular property for sublinear expectations, we get a strong one. These results are helpful for the application of the law of large numbers in engineering and statistics such as to assess the level of product quality and to explain the relationship between the statistical distribution and the sample distribution."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.05631","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1511.05631/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T07:13:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I2mkhtt2xE8amiMzfDTabvdZTETQXvmGAutaWm82gqpkJ9stWdBw2AVvUf0X/muskj6yq+++viSPgmi9aTIRCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T11:08:20.446523Z"},"content_sha256":"06b35304e05c7eb803dc80b6661a05b34ab4896783586e783486ebfd1dde95d9","schema_version":"1.0","event_id":"sha256:06b35304e05c7eb803dc80b6661a05b34ab4896783586e783486ebfd1dde95d9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2Z42RRBQT7YEEWBH6HQWM3OZID/bundle.json","state_url":"https://pith.science/pith/2Z42RRBQT7YEEWBH6HQWM3OZID/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2Z42RRBQT7YEEWBH6HQWM3OZID/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":"2026-08-13T11:08:20Z","links":{"resolver":"https://pith.science/pith/2Z42RRBQT7YEEWBH6HQWM3OZID","bundle":"https://pith.science/pith/2Z42RRBQT7YEEWBH6HQWM3OZID/bundle.json","state":"https://pith.science/pith/2Z42RRBQT7YEEWBH6HQWM3OZID/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2Z42RRBQT7YEEWBH6HQWM3OZID/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:2Z42RRBQT7YEEWBH6HQWM3OZID","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":"08dfa7737d3be8d378d71010b902204977d895fcc69af5d26ba4f302bb755bd3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-11-18T00:52:20Z","title_canon_sha256":"729b07b3d559a77f71fb6c902d2c6f2eec58456bfb0491025886a419493a7d08"},"schema_version":"1.0","source":{"id":"1511.05631","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.05631","created_at":"2026-07-05T07:13:17Z"},{"alias_kind":"arxiv_version","alias_value":"1511.05631v2","created_at":"2026-07-05T07:13:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.05631","created_at":"2026-07-05T07:13:17Z"},{"alias_kind":"pith_short_12","alias_value":"2Z42RRBQT7YE","created_at":"2026-07-05T07:13:17Z"},{"alias_kind":"pith_short_16","alias_value":"2Z42RRBQT7YEEWBH","created_at":"2026-07-05T07:13:17Z"},{"alias_kind":"pith_short_8","alias_value":"2Z42RRBQ","created_at":"2026-07-05T07:13:17Z"}],"graph_snapshots":[{"event_id":"sha256:06b35304e05c7eb803dc80b6661a05b34ab4896783586e783486ebfd1dde95d9","target":"graph","created_at":"2026-07-05T07:13:17Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1511.05631/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we consider the sublinear expectation on bounded random variables. With the notion of uncorrelatedness for random variables under the sublinear expectation, a weak law of large numbers is obtained. With the notion of independence for random variable sequences and regular property for sublinear expectations, we get a strong one. These results are helpful for the application of the law of large numbers in engineering and statistics such as to assess the level of product quality and to explain the relationship between the statistical distribution and the sample distribution.","authors_text":"Chuanfeng Sun, Wenhao Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-11-18T00:52:20Z","title":"Two laws of large numbers for sublinear expectations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.05631","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:fa58b5ece569194663fe1bb212e082208efcf5f7928cbbd19a1779ba29fb55f2","target":"record","created_at":"2026-07-05T07:13:17Z","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":"08dfa7737d3be8d378d71010b902204977d895fcc69af5d26ba4f302bb755bd3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-11-18T00:52:20Z","title_canon_sha256":"729b07b3d559a77f71fb6c902d2c6f2eec58456bfb0491025886a419493a7d08"},"schema_version":"1.0","source":{"id":"1511.05631","kind":"arxiv","version":2}},"canonical_sha256":"d679a8c4309ff0425827f1e1666dd940d6ffd9d2a2949d3805d64c620320423a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d679a8c4309ff0425827f1e1666dd940d6ffd9d2a2949d3805d64c620320423a","first_computed_at":"2026-07-05T07:13:17.622148Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:13:17.622148Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"P6aQizlh8KqWy/IV5wk1jH/oVxSVnA6Vqy4/nlgo6OpZurbNmxNMarxP/JYWdlBBiDCzNTfW1kwEOeq/m85tDw==","signature_status":"signed_v1","signed_at":"2026-07-05T07:13:17.622574Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.05631","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fa58b5ece569194663fe1bb212e082208efcf5f7928cbbd19a1779ba29fb55f2","sha256:06b35304e05c7eb803dc80b6661a05b34ab4896783586e783486ebfd1dde95d9"],"state_sha256":"75161c0f69069f4cb1bdb3e11fff0ffee8f2763e38caff3b9b83df0ca07c17e6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qQxMr9MqrHLnS9JbVWRPVaKe+o96F1rK5UjBoY1AfsDrIL1FP5CLBDeA+505iCF3C2iR2n0D+8hEr/v1M/GUBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T11:08:20.451450Z","bundle_sha256":"3a8e628cf2b29a229b302f899705a63cb7c3b40ab3d6bf534f0e0d779991e7cd"}}