{"paper":{"title":"Variation inequalities for smartingales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Markus Passenbrunner","submitted_at":"2024-09-20T05:22:52Z","abstract_excerpt":"A result by N.G. Makarov [Algebra i Analiz, 1989] states that for martingales $(M_n)$ on the torus we have the strict inequality \\[ \\liminf_{n\\to\\infty} \\frac{M_n}{\\sum_{k=1}^n |\\Delta M_k|} > 0 \\] on a set of Hausdorff dimension one, denoting by $\\Delta M_n$ the martingale differences $ \\Delta M_n = M_n - M_{n-1} $. We discuss an extension of this inequality to so-called smartingales on convex, compact subsets of $\\mathbb R^d$, which are piecewise polynomial (or spline) versions of martingales. As a tool we need and prove an estimate for smartingales in the spirit of the law of the iterated l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.13227","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2409.13227/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"}