{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:IP7ZILYTHRW4UCD5DQ2LFC7AFR","short_pith_number":"pith:IP7ZILYT","schema_version":"1.0","canonical_sha256":"43ff942f133c6dca087d1c34b28be02c586628fa2d4a04e135c8eacbf2e0967e","source":{"kind":"arxiv","id":"0907.4450","version":3},"attestation_state":"computed","paper":{"title":"Nonnormal approximation by Stein's method of exchangeable pairs with application to the Curie--Weiss model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Qi-Man Shao, Sourav Chatterjee","submitted_at":"2009-07-25T22:29:23Z","abstract_excerpt":"Let $(W,W')$ be an exchangeable pair. Assume that \\[E(W-W'|W)=g(W)+r(W),\\] where $g(W)$ is a dominated term and $r(W)$ is negligible. Let $G(t)=\\int_0^tg(s)\\,ds$ and define $p(t)=c_1e^{-c_0G(t)}$, where $c_0$ is a properly chosen constant and $c_1=1/\\int_{-\\infty}^{\\infty}e^{-c_0G(t)}\\,dt$. Let $Y$ be a random variable with the probability density function $p$. It is proved that $W$ converges to $Y$ in distribution when the conditional second moment of $(W-W')$ given $W$ satisfies a law of large numbers. A Berry-Esseen type bound is also given. We use this technique to obtain a Berry-Esseen er"},"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":"0907.4450","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-07-25T22:29:23Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"cfa231dffb320eae4384880beb1c3cde2388c82f0296fc9f4839f2dec6235095","abstract_canon_sha256":"bd4ef9690ad45e9064cfaa97f87aee3b14d815f78e965eeda2991176e357ec9a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:24:36.397344Z","signature_b64":"MBGWqckQOK4mSYjYGzQjm/6G/inARcbyotGKmapcXSlb45CGlzfezGGHE7pWBjwOaJjlK5fy/wEFubp02sBdAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"43ff942f133c6dca087d1c34b28be02c586628fa2d4a04e135c8eacbf2e0967e","last_reissued_at":"2026-05-18T04:24:36.396895Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:24:36.396895Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nonnormal approximation by Stein's method of exchangeable pairs with application to the Curie--Weiss model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Qi-Man Shao, Sourav Chatterjee","submitted_at":"2009-07-25T22:29:23Z","abstract_excerpt":"Let $(W,W')$ be an exchangeable pair. Assume that \\[E(W-W'|W)=g(W)+r(W),\\] where $g(W)$ is a dominated term and $r(W)$ is negligible. Let $G(t)=\\int_0^tg(s)\\,ds$ and define $p(t)=c_1e^{-c_0G(t)}$, where $c_0$ is a properly chosen constant and $c_1=1/\\int_{-\\infty}^{\\infty}e^{-c_0G(t)}\\,dt$. Let $Y$ be a random variable with the probability density function $p$. It is proved that $W$ converges to $Y$ in distribution when the conditional second moment of $(W-W')$ given $W$ satisfies a law of large numbers. A Berry-Esseen type bound is also given. 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