{"paper":{"title":"The value at the mode in multivariate $t$ distributions: a curiosity or not?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR","stat.TH"],"primary_cat":"math.ST","authors_text":"Anouk Neven, Christophe Ley","submitted_at":"2012-11-06T10:49:04Z","abstract_excerpt":"It is a well-known fact that multivariate Student $t$ distributions converge to multivariate Gaussian distributions as the number of degrees of freedom $\\nu$ tends to infinity, irrespective of the dimension $k\\geq1$. In particular, the Student's value at the mode (that is, the normalizing constant obtained by evaluating the density at the center) $c_{\\nu,k}=\\frac{\\Gamma(\\frac{\\nu+k}{2})}{(\\pi \\nu)^{k/2} \\Gamma( \\frac{\\nu}{2})}$ converges towards the Gaussian value at the mode $c_k=\\frac{1}{(2\\pi)^{k/2}}$. In this note, we prove a curious fact: $c_{\\nu,k}$ tends monotonically to $c_k$ for each "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.1174","kind":"arxiv","version":3},"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"}