{"paper":{"title":"Characterization problems for linear forms with free summands","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"F. G\\\"otze, G. P. Chistyakov","submitted_at":"2011-10-07T13:51:24Z","abstract_excerpt":"Let $T_1,...,T_n$ denote free random variables. For two linear forms $L_1=\\sum_{j=1}^n a_jT_j$ and $L_2=\\sum_{j=1}^n b_jT_j$ with real coefficients $a_j$ and $b_j$ we shall describe all distributions of $T_1,...,T_n$ such that $L_1$ and $L_2$ are free. For identically distributed free random variables $T_1,...,T_n$ with distribution $\\mu$ we establish necessary and sufficient conditions on the coefficients $a_j,b_j,\\,j=1,...,n,$ such that the statements:\\quad $(i)$ $\\mu$ is a centered semicircular distribution; and $(ii)$ \\, $L_1$ and $L_2$ are identically distributed ($L_1\\stackrel{D}{=}L_2$)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1527","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"}