{"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"}