{"paper":{"title":"Self-normalized deviation inequalities with application to $t$-statistics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Xiequan Fan","submitted_at":"2016-11-25T12:30:39Z","abstract_excerpt":"Let $(\\xi_i)_{i=1,...,n}$ be a sequence of independent and symmetric random variables. We consider the upper bounds on tail probabilities of self-normalized deviations $$ \\mathbf{P} \\Big( \\max_{1\\leq k \\leq n} \\sum_{i=1}^{k} |\\xi_i|\\big/ \\big(\\sum_{i=1}^{n} |\\xi_i|^\\beta \\big)^{1/\\beta} \\geq x \\Big) $$ for $x>0$ and $\\beta >1.$ Our bound is the best that can be obtained from the Bernstein inequality under the present assumption. An application to Student's $t$-statistics is also given."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.08436","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"}