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Our main result is the following general bound: there exist two finite constants c,C>0 such that, for n sufficiently large, c max(|E[F_n^3]|, E[F_n^4]-3) < d(F_n,N) < C max(|E[F_n^3]|, E[F_n^4]-3), where d(F_n,N) = sup |E[h(F_n)] - E[h(N)]|, and h runs over the class of all real functions with a second derivative bounded by 1. This shows that the deterministic sequence max(|E[F_n^3]|, 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":"1109.1546","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-07T19:42:49Z","cross_cats_sorted":[],"title_canon_sha256":"812e174b6075dd977940963cff7b310397fe11da4cdfff676f30f1025fe463e4","abstract_canon_sha256":"fc43b965c6147e1a9d80c3d940739658cb5423ede4e84d0f79f5d9efa7a8a020"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:13:55.498383Z","signature_b64":"G8qbCdOk/tKlKs5GwygixHN3wiMi6Sg6PDF9USm8fKHSGgkVRSkpeV6p8sctseM4n2EialyH5s8JmouoeIODCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c316d2f731ef821da49982e937f659432b2c4db9b33261fb171d9c38ac333a7b","last_reissued_at":"2026-05-18T04:13:55.497876Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:13:55.497876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Aline Bonami (MAPMO), Giovanni Peccati, Hermine Bierm\\'e (MAP5, Ivan Nourdin (IECN), LPMT)","submitted_at":"2011-09-07T19:42:49Z","abstract_excerpt":"Let {F_n} be a normalized sequence of random variables in some fixed Wiener chaos associated with a general Gaussian field, and assume that E[F_n^4] --> E[N^4]=3, where N is a standard Gaussian random variable. Our main result is the following general bound: there exist two finite constants c,C>0 such that, for n sufficiently large, c max(|E[F_n^3]|, E[F_n^4]-3) < d(F_n,N) < C max(|E[F_n^3]|, E[F_n^4]-3), where d(F_n,N) = sup |E[h(F_n)] - E[h(N)]|, and h runs over the class of all real functions with a second derivative bounded by 1. 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