{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:H5FSYKCOTIA7PP6WZVPYOQD7HD","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":"dfcf45d63879ce48354111f18bcc58225e8cf6eb4dacd7d0fc416455b8a0bc22","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-11-15T02:27:44Z","title_canon_sha256":"76a27706fd537643192558acfe4577015d50480432179741078be2c8bd49c751"},"schema_version":"1.0","source":{"id":"1711.05384","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.05384","created_at":"2026-05-18T00:30:33Z"},{"alias_kind":"arxiv_version","alias_value":"1711.05384v1","created_at":"2026-05-18T00:30:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.05384","created_at":"2026-05-18T00:30:33Z"},{"alias_kind":"pith_short_12","alias_value":"H5FSYKCOTIA7","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_16","alias_value":"H5FSYKCOTIA7PP6W","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_8","alias_value":"H5FSYKCO","created_at":"2026-05-18T12:31:18Z"}],"graph_snapshots":[{"event_id":"sha256:54872812a0929df3d4edfec592b38adc0b1a78043cc98e6f03644fb688298c91","target":"graph","created_at":"2026-05-18T00:30:33Z","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":"Peng (2008)(\\cite{P08b}) proved the Central Limit Theorem under a sublinear expectation:\n  \\textit{Let $(X_i)_{i\\ge 1}$ be a sequence of i.i.d random variables under a sublinear expectation $\\hat{\\mathbf{E}}$ with $\\hat{\\mathbf{E}}[X_1]=\\hat{\\mathbf{E}}[-X_1]=0$ and $\\hat{\\mathbf{E}}[|X_1|^3]<\\infty$. Setting $W_n:=\\frac{X_1+\\cdots+X_n}{\\sqrt{n}}$, we have, for each bounded and Lipschitz function $\\varphi$, \\[\\lim_{n\\rightarrow\\infty}\\bigg|\\hat{\\mathbf{E}}[\\varphi(W_n)]-\\mathcal{N}_G(\\varphi)\\bigg|=0,\\] where $\\mathcal{N}_G$ is the $G$-normal distribution with $G(a)=\\frac{1}{2}\\hat{\\mathbf{E}}","authors_text":"Yongsheng Song","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-11-15T02:27:44Z","title":"Normal Approximation by Stein's Method under Sublinear Expectations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.05384","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":"sha256:6237a54940b35f402473a394daa38bd5957781d48da962b065f78de2c4164633","target":"record","created_at":"2026-05-18T00:30:33Z","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":"dfcf45d63879ce48354111f18bcc58225e8cf6eb4dacd7d0fc416455b8a0bc22","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-11-15T02:27:44Z","title_canon_sha256":"76a27706fd537643192558acfe4577015d50480432179741078be2c8bd49c751"},"schema_version":"1.0","source":{"id":"1711.05384","kind":"arxiv","version":1}},"canonical_sha256":"3f4b2c284e9a01f7bfd6cd5f87407f38c5d20ec638b0e623805e5b3a640d357f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3f4b2c284e9a01f7bfd6cd5f87407f38c5d20ec638b0e623805e5b3a640d357f","first_computed_at":"2026-05-18T00:30:33.436665Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:30:33.436665Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KuIO0SthrGLDUSLlveXxPDZ1Tmxj99KNpwpIFZQiJTsbbArG9gLyW0vSsfOp2cIY0MVifn6zk4pmGtHJlvB5DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:30:33.437369Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.05384","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6237a54940b35f402473a394daa38bd5957781d48da962b065f78de2c4164633","sha256:54872812a0929df3d4edfec592b38adc0b1a78043cc98e6f03644fb688298c91"],"state_sha256":"f94bb560c2324ede2c89ebf7f2b1f590c4eab53e7ec41e8dc04bba4329a12664"}