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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}}"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1711.05384","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-11-15T02:27:44Z","cross_cats_sorted":[],"title_canon_sha256":"76a27706fd537643192558acfe4577015d50480432179741078be2c8bd49c751","abstract_canon_sha256":"dfcf45d63879ce48354111f18bcc58225e8cf6eb4dacd7d0fc416455b8a0bc22"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:33.437369Z","signature_b64":"KuIO0SthrGLDUSLlveXxPDZ1Tmxj99KNpwpIFZQiJTsbbArG9gLyW0vSsfOp2cIY0MVifn6zk4pmGtHJlvB5DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f4b2c284e9a01f7bfd6cd5f87407f38c5d20ec638b0e623805e5b3a640d357f","last_reissued_at":"2026-05-18T00:30:33.436665Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:33.436665Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Normal Approximation by Stein's Method under Sublinear Expectations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Yongsheng Song","submitted_at":"2017-11-15T02:27:44Z","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}}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.05384","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1711.05384","created_at":"2026-05-18T00:30:33.436778+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.05384v1","created_at":"2026-05-18T00:30:33.436778+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.05384","created_at":"2026-05-18T00:30:33.436778+00:00"},{"alias_kind":"pith_short_12","alias_value":"H5FSYKCOTIA7","created_at":"2026-05-18T12:31:18.294218+00:00"},{"alias_kind":"pith_short_16","alias_value":"H5FSYKCOTIA7PP6W","created_at":"2026-05-18T12:31:18.294218+00:00"},{"alias_kind":"pith_short_8","alias_value":"H5FSYKCO","created_at":"2026-05-18T12:31:18.294218+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/H5FSYKCOTIA7PP6WZVPYOQD7HD","json":"https://pith.science/pith/H5FSYKCOTIA7PP6WZVPYOQD7HD.json","graph_json":"https://pith.science/api/pith-number/H5FSYKCOTIA7PP6WZVPYOQD7HD/graph.json","events_json":"https://pith.science/api/pith-number/H5FSYKCOTIA7PP6WZVPYOQD7HD/events.json","paper":"https://pith.science/paper/H5FSYKCO"},"agent_actions":{"view_html":"https://pith.science/pith/H5FSYKCOTIA7PP6WZVPYOQD7HD","download_json":"https://pith.science/pith/H5FSYKCOTIA7PP6WZVPYOQD7HD.json","view_paper":"https://pith.science/paper/H5FSYKCO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.05384&json=true","fetch_graph":"https://pith.science/api/pith-number/H5FSYKCOTIA7PP6WZVPYOQD7HD/graph.json","fetch_events":"https://pith.science/api/pith-number/H5FSYKCOTIA7PP6WZVPYOQD7HD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/H5FSYKCOTIA7PP6WZVPYOQD7HD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/H5FSYKCOTIA7PP6WZVPYOQD7HD/action/storage_attestation","attest_author":"https://pith.science/pith/H5FSYKCOTIA7PP6WZVPYOQD7HD/action/author_attestation","sign_citation":"https://pith.science/pith/H5FSYKCOTIA7PP6WZVPYOQD7HD/action/citation_signature","submit_replication":"https://pith.science/pith/H5FSYKCOTIA7PP6WZVPYOQD7HD/action/replication_record"}},"created_at":"2026-05-18T00:30:33.436778+00:00","updated_at":"2026-05-18T00:30:33.436778+00:00"}