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Replacing the quadratic characteristic by $\\textrm{H}_k^y:= \\sum_{i=1}^k\\left(\\mathbf{E}(\\xi_i^2 |\\mathcal{F}_{i-1}) +\\xi_i^2\\textbf{1}_{\\{|\\xi_i|> y\\}}\\right),$ Dzhaparidze and van Zanten (\\emph{Stochastic Process. Appl.}, 2001) have extended Freedman's inequality to martingales with unbounded differences. In this paper, we prove tha"},"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":"1404.4776","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-04-18T12:55:25Z","cross_cats_sorted":[],"title_canon_sha256":"73b1f0a581374ea13193ec36ec70123f5d9571451ed1b9a1226e2ee6919ff88a","abstract_canon_sha256":"66dd7d0fb8d92bcd9f3e5c0692b3172762609ce9128a5e437505ce180f2065fb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:46.961489Z","signature_b64":"Z5AnjAcxM+beyGmBbfvvSy1DF2fO2LUDNnmG7/GYzZeNTGb5egrs6fqvui5xDJp8UPzMkx8YgU+kZT92thY4AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"605e64cce2106690cded8310c0a658f6008ce70478e318367aca5423b5a970f2","last_reissued_at":"2026-05-18T00:38:46.960858Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:46.960858Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Martingale inequalities of type Dzhaparidze and van Zanten","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":"2014-04-18T12:55:25Z","abstract_excerpt":"Freedman's inequality is a supermartingale counterpart to Bennett's inequality. This result shows that the tail probabilities of a supermartingale is controlled by the quadratic characteristic and a uniform upper bound for the supermartingale difference sequence. Replacing the quadratic characteristic by $\\textrm{H}_k^y:= \\sum_{i=1}^k\\left(\\mathbf{E}(\\xi_i^2 |\\mathcal{F}_{i-1}) +\\xi_i^2\\textbf{1}_{\\{|\\xi_i|> y\\}}\\right),$ Dzhaparidze and van Zanten (\\emph{Stochastic Process. Appl.}, 2001) have extended Freedman's inequality to martingales with unbounded differences. 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