{"paper":{"title":"Harmonic Means of Wishart Random Matrices","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Asad Lodhia","submitted_at":"2019-05-07T05:10:32Z","abstract_excerpt":"We use free probability to compute the limiting spectral properties of the harmonic mean of $n$ i.i.d. Wishart random matrices $\\mathbf{W}_i$ whose limiting aspect ratio is $\\gamma \\in (0,1)$ when $\\mathbb{E}[\\mathbf{W}_i] = \\mathbf{I}$. We demonstrate an interesting phenomenon where the harmonic mean $\\mathbf{H}$ of the $n$ Wishart matrices is closer in operator norm to $\\mathbb{E}[\\mathbf{W}_i]$ than the arithmetic mean $\\mathbf{A}$ for small $n$, after which the arithmetic mean is closer. We also prove some results for the general case where the expectation of the Wishart matrices are not t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.02357","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":""},"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"}