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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}."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1111.1407","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-11-06T11:37:03Z","cross_cats_sorted":[],"title_canon_sha256":"490e8e4a908f42d44464e84a7d81e448a06c9c29196091adbad5ae63d792cd80","abstract_canon_sha256":"c5a8bf38e8377e88802f9e4e364b5422969cad1a4125bed39ac8435c8a1dcf78"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:32.621207Z","signature_b64":"d9rslN5Vp7mVMoqEWGzmYW8qolT7CkSgP35MbuNgnIsLuJUVNDqxOBKw2wHH6X2mwyh7DQNpgFe8FG+pnwg7BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06843860209f3d14438ba4f5773b63e1f6d5782055fbfd50c2282cec974ef2c4","last_reissued_at":"2026-05-18T03:26:32.620854Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:32.620854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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