{"paper":{"title":"On the heavy-tailedness of Student's $t$-statistic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Fredrik Jonsson","submitted_at":"2011-02-10T10:25:27Z","abstract_excerpt":"Let $\\{X_i\\}_{i\\geq1}$ be an i.i.d. sequence of random variables and define, for $n\\geq2$, \\[T_n=\\cases{n^{-1/2}\\hat{\\sigma}_n^{-1}S_n,\\quad \\hat{\\sigma}_n>0,\\cr 0,\\quad \\hat{\\sigma}_n=0,}with S_n=\\sum_{i=1}^nX_i, \\hat{\\sigma}^2_n=\\frac{1}{n-1}\\sum_{i=1}^n(X_i-n^{-1}S_n)^2.\\] We investigate the connection between the distribution of an observation $X_i$ and finiteness of $\\mathrm{E}|T_n|^r$ for $(n,r)\\in \\mathbb{N}_{\\geq2}\\times\\mathbb{R}^+$. Moreover, assuming $T_n\\stackrel{d}{\\longrightarrow}T$, we prove that for any $r>0$, $\\lim_{n\\to\\infty}\\mathrm{E}|T_n|^r=\\mathrm{E}|T|^r<\\infty$, provide"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.2072","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"}