{"paper":{"title":"Sharp maximal inequalities for the moments of martingales and non-negative submartingales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Adam Os\\c{e}kowski","submitted_at":"2012-01-05T09:18:24Z","abstract_excerpt":"In the paper we study sharp maximal inequalities for martingales and non-negative submartingales: if $f$, $g$ are martingales satisfying \\[|\\mathrm{d}g_n|\\leq|\\mathrm{d}f_n|,\\qquad n=0,1,2,...,\\] almost surely, then \\[\\Bigl\\|\\sup_{n\\geq0}|g_n|\\Bigr\\|_p\\leq p\\|f\\|_p,\\qquad p\\geq2,\\] and the inequality is sharp. Furthermore, if $\\alpha\\in[0,1]$, $f$ is a non-negative submartingale and $g$ satisfies \\[|\\mathrm{d}g_n|\\leq|\\mathrm{d}f_n|\\quad and\\quad |\\mathbb{E}(\\mathrm{d}g_{n+1}|\\mathcal {F}_n)|\\leq\\alpha\\mathbb{E}(\\mathrm{d}f_{n+1}|\\mathcal{F}_n),\\qquad n=0,1,2,...,\\] almost surely, then \\[\\Bigl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.1089","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}