{"paper":{"title":"Uniform oscillatory behavior of spherical functions of $GL_n/U_n$ at the identity and a central limit theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CA","authors_text":"Michael Voit","submitted_at":"2012-05-22T10:12:42Z","abstract_excerpt":"Let $\\mathbb F=\\mathbb R$ or $\\mathbb C$ and $n\\in\\b N$.\n  Let $(S_k)_{k\\ge0}$ be a time-homogeneous random walk on $GL_n(\\b F)$ associated with an $U_n(\\b F)$-biinvariant measure $\\nu\\in M^1(GL_n(\\b F))$. We derive a central limit theorem for the ordered singular spectrum $\\sigma_{sing}(S_k)$ with a normal distribution as limit with explicit analytic formulas for the drift vector and the covariance matrix.\n  The main ingredient for the proof will be a oscillatory result for the spherical functions $\\phi_{i\\rho+\\lambda}$ of $(GL_n(\\b F),U_n(\\b F))$. More precisely, we present a necessarily uni"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4866","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"}