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In this report, we follow Barron's steps as we prove the existence of $J_{\\alpha}\\left(X + \\sqrt[\\alpha]{\\eta}N\\right)$, $\\eta > 0$ for any Radom Variable (RV) $X \\in \\mathcal{L}$ where \\begin{equation*} \\mathcal{L} = \\left\\{ \\text{RVs} \\,\\,U: \\int \\ln\\left(1 + |U|\\right)\\,dF_{U}(u) \\text{ is finite } \\right\\}, \\end{equation*} and where $N \\sim \\mathcal{S}(\\alpha;1)$ is independent of $X$, $0< \\alpha <2$."},"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":"1604.02058","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-04-07T16:14:39Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"185f243308addddc07f9f9844ca4b72416c9e84a38f374e087fa2b59f7d275f7","abstract_canon_sha256":"5e27d22b88cd02d8c4f3c2ddddce15baa4dcf51feb9c618809fe8faec321e743"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:31.346853Z","signature_b64":"ASZtXPXCYaJVZr7xkqc3jtIbzZW3bAAY0/nvPS4c8fh+OcmdUaT9fMz61cdnQD/66amHr2ST2OD9LxmbfdarBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cbc2e48b028151d0efffca8434c9eb4a673fc92af7b938d7461c25d6904e1757","last_reissued_at":"2026-05-18T01:17:31.346157Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:31.346157Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sufficient Conditions for Existence of $J_{\\alpha}(X + \\sqrt[\\alpha]{\\eta}N)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Ibrahim Abou-Faycal, Jihad Fahs","submitted_at":"2016-04-07T16:14:39Z","abstract_excerpt":"In his technical report~\\cite[sec. 6]{barrontech}, Barron states that the de Bruijn's identity for Gaussian perturbations holds for any RV having a finite variance. 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