{"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. 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 $X$, $0< \\alpha <2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02058","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"}