{"paper":{"title":"Large deviation exponential inequalities for supermartingales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ion Grama, Quansheng Liu, Xiequan Fan","submitted_at":"2011-11-06T11:37:03Z","abstract_excerpt":"Let $(X_{i}, \\mathcal{F}_{i})_{i\\geq 1}$ be a sequence of supermartingale differences and let $S_k=\\sum_{i=1}^k X_i$. We give an exponential moment condition under which $P(\\max_{1\\leq k \\leq n} S_k \\geq n)=O(\\exp\\{-C_1 n^{\\alpha}\\}),$ $n\\rightarrow \\infty,$ where $\\alpha \\in (0, 1)$ is given and $C_{1}>0$ is a constant. We also show that the power $\\alpha$ is optimal under the given condition. In particular, when $\\alpha=1/3$, we recover an inequality of Lesigne and Voln\\'{y}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.1407","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"}