{"paper":{"title":"Classical and free Fourth Moment Theorems: universality and thresholds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"DiMIE), Giovanni Peccati (FSTC), Guillaume Poly (FSTC), Ivan Nourdin (FSTC), Rosaria Simone (FSTC","submitted_at":"2014-07-23T13:56:09Z","abstract_excerpt":"Let $X$ be a centered random variable with unit variance, zero third moment, and such that $E[X^4] \\ge 3$. Let $\\{F_n : n\\geq 1\\}$ denote a normalized sequence of homogeneous sums of fixed degree $d\\geq 2$, built from independent copies of $X$. Under these minimal conditions, we prove that $F_n$ converges in distribution to a standard Gaussian random variable if and only if the corresponding sequence of fourth moments converges to $3$. The statement is then extended (mutatis mutandis) to the free probability setting. We shall also discuss the optimality of our conditions in terms of explicit t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6216","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"}