{"paper":{"title":"Self-normalized Cram\\'{e}r type moderate deviations for the maximum of sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Qi-Man Shao, Qiying Wang, Weidong Liu","submitted_at":"2013-07-23T12:41:33Z","abstract_excerpt":"Let $X_1,X_2,...$ be independent random variables with zero means and finite variances, and let $S_n=\\sum_{i=1}^nX_i$ and $V^2_n=\\sum_{i=1}^nX^2_i$. A Cram\\'{e}r type moderate deviation for the maximum of the self-normalized sums $\\max_{1\\leq k\\leq n}S_k/V_n$ is obtained. In particular, for identically distributed $X_1,X_2,...,$ it is proved that $P(\\max_{1\\leq k\\leq n}S_k\\geq xV_n)/(1-\\Phi (x))\\rightarrow2$ uniformly for $0<x\\leq\\mathrm{o}(n^{1/6})$ under the optimal finite third moment of $X_1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.6044","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"}