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Let $\\mu$ be a probability distribution on $X$ such that its characteristic function $\\hat\\mu(y)$ does not vanish and $\\hat\\mu(y)$ for some $n \\geq 3$ satisfies the equation $$ \\prod_{j=1}^{n} \\hat\\mu(y_j + y) = \\prod_{j=1}^{n}\\hat\\mu(y_j - y), \\quad \\sum_{j=1}^{n} y_j = 0, \\quad y_1,\\dots,y_n,y \\in Y. $$ Then $\\mu$ is a convolution of a Gaussian distribution and a distribution supported in the subgroup of $X$ generated by elements of order 2."},"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":"1309.6770","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-09-26T09:27:15Z","cross_cats_sorted":[],"title_canon_sha256":"58537bd06c25c3994086617de315dea26366fe21df2bc81527b6eb8da8418266","abstract_canon_sha256":"8380b9e31e4e0877c0da268a20c95a56bb96d75229d104624e42b2005215b289"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:10.585013Z","signature_b64":"lMhivNMK0lTU2hToULf8xbSjP5DkgJbBYSw65eBWuEgVPoMpC8NbDMCRlEHBmc6n5B0OL6TQEeQmW+I67OHeCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f52e78aa8ba3a3329e8d004db6729de2dcd4da46c115f2dade0ba22e6130629","last_reissued_at":"2026-05-18T03:12:10.584185Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:10.584185Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a functional equation appearing in characterization of distributions by the optimality of an estimate","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"G. 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