{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:A2CDQYBAT46RIQ4LUT2XOO3D4H","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c5a8bf38e8377e88802f9e4e364b5422969cad1a4125bed39ac8435c8a1dcf78","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-11-06T11:37:03Z","title_canon_sha256":"490e8e4a908f42d44464e84a7d81e448a06c9c29196091adbad5ae63d792cd80"},"schema_version":"1.0","source":{"id":"1111.1407","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.1407","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"arxiv_version","alias_value":"1111.1407v3","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.1407","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"pith_short_12","alias_value":"A2CDQYBAT46R","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"A2CDQYBAT46RIQ4L","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"A2CDQYBA","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:65873151f6dec25a20dad8be65896eca5ae2850781d5910c2808a71885427f07","target":"graph","created_at":"2026-05-18T03:26:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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}.","authors_text":"Ion Grama, Quansheng Liu, Xiequan Fan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-11-06T11:37:03Z","title":"Large deviation exponential inequalities for supermartingales"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.1407","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d4192dbed25096729ed10d2d2ede2d9a0ff25965d998db6a184a08892b65481c","target":"record","created_at":"2026-05-18T03:26:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c5a8bf38e8377e88802f9e4e364b5422969cad1a4125bed39ac8435c8a1dcf78","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-11-06T11:37:03Z","title_canon_sha256":"490e8e4a908f42d44464e84a7d81e448a06c9c29196091adbad5ae63d792cd80"},"schema_version":"1.0","source":{"id":"1111.1407","kind":"arxiv","version":3}},"canonical_sha256":"06843860209f3d14438ba4f5773b63e1f6d5782055fbfd50c2282cec974ef2c4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"06843860209f3d14438ba4f5773b63e1f6d5782055fbfd50c2282cec974ef2c4","first_computed_at":"2026-05-18T03:26:32.620854Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:26:32.620854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"d9rslN5Vp7mVMoqEWGzmYW8qolT7CkSgP35MbuNgnIsLuJUVNDqxOBKw2wHH6X2mwyh7DQNpgFe8FG+pnwg7BA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:26:32.621207Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.1407","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d4192dbed25096729ed10d2d2ede2d9a0ff25965d998db6a184a08892b65481c","sha256:65873151f6dec25a20dad8be65896eca5ae2850781d5910c2808a71885427f07"],"state_sha256":"8b1fa530b3cac0f0578b064d8f2c07d1d8b9500f243f155ce58581aa09445dc4"}