{"paper":{"title":"Rates of convergence in the Free Multiplicative Central Limit Theorem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.OA","authors_text":"Marwa Banna, Nicolas Gilliers, Pei-Lun Tseng","submitted_at":"2025-05-23T20:09:51Z","abstract_excerpt":"We provide the first quantitative estimates for the rate of convergence in the free multiplicative central limit theorem (CLT), in terms of the Kolmogorov and $r$-Wasserstein distances for $r \\geq 1$. While the free additive CLT has been thoroughly studied, including convergence rates, the multiplicative setting remained open in this regard. We consider products of the form $$ \\pi_n^{g,n^{-1/2}x} := g\\left(\\frac{x_1}{\\sqrt{n}}\\right) \\cdots g\\left(\\frac{x_n}{\\sqrt{n}}\\right),$$ where $x_1, \\dots, x_n$ are freely independent self-adjoint operators with common variance $\\sigma^2$ and $g \\colon \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.18348","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.18348/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}