{"paper":{"title":"From a stochastic maximal inequality to infinite-dimensional martingales, towards high-dimensional statistics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A novel oracle maximal inequality via integration by parts yields sharp bounds for martingale random field suprema.","cross_cats":[],"primary_cat":"math.PR","authors_text":"Yoichi Nishiyama","submitted_at":"2026-03-31T13:37:15Z","abstract_excerpt":"A novel approach is proposed to establish a sharp upper bound on the expected supremum of a separable martingale random field, serving as an alternative to classical universal chaining-based methods. The proposed approach begins by deriving a new \"stochastic maximal inequality\" for a finite class of discrete-time martingales. This is achieved by using some variations of log-sum-exp and softmax functions, as well as martingale transforms, avoiding the simple use of the triangle inequalty. We apply this inequality to obtain a generalization of Lenglart's inequality for discrete-time martingales,"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"A novel approach is proposed to establish a sharp upper bound on the expected supremum of a separable martingale random field, serving as an alternative to classical universal chaining-based methods. This yields a generalization of Lenglart's inequality to finite-dimensional and certain infinite-dimensional settings via a finite approximation device.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The finite approximation device successfully extends the finite-dimensional generalization to the relevant infinite-dimensional cases while preserving the sharpness of the bound, assuming the martingale random field is separable.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A new oracle maximal inequality for finite submartingales is derived via integration by parts, generalizing Lenglart's inequality to finite- and certain infinite-dimensional martingale random fields through finite approximation from below.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A novel oracle maximal inequality via integration by parts yields sharp bounds for martingale random field suprema.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d658f3aaa113d2610e34c57222d621f3217c2b1e52bf8f403a17544af89014e5"},"source":{"id":"2603.29739","kind":"arxiv","version":3},"verdict":{"id":"6d75976e-0890-46ad-9361-86a61773aadc","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T23:29:51.735533Z","strongest_claim":"A novel approach is proposed to establish a sharp upper bound on the expected supremum of a separable martingale random field, serving as an alternative to classical universal chaining-based methods. This yields a generalization of Lenglart's inequality to finite-dimensional and certain infinite-dimensional settings via a finite approximation device.","one_line_summary":"A new oracle maximal inequality for finite submartingales is derived via integration by parts, generalizing Lenglart's inequality to finite- and certain infinite-dimensional martingale random fields through finite approximation from below.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The finite approximation device successfully extends the finite-dimensional generalization to the relevant infinite-dimensional cases while preserving the sharpness of the bound, assuming the martingale random field is separable.","pith_extraction_headline":"A novel oracle maximal inequality via integration by parts yields sharp bounds for martingale random field suprema."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.29739/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}