{"paper":{"title":"Maximal generalization of Baum-Katz theorem and optimality of sequential tests","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Didier Piau","submitted_at":"2005-10-03T12:49:06Z","abstract_excerpt":"Baum-Katz theorem asserts that the Cesaro means of i.i.d. increments distributed like X r-converge if and only if |X|^{r+1} is integrable. We generalize this, and we unify other results, by proving that the following equivalence holds, if and only if G is moderate: the Cesaro means G-converge if and only if G(L(a)) is integrable for every a if and only if |X|.G(|X|) is integrable. Here, L(a) is the last time when the deviation of the Cesaro mean from its limit exceeds a, and G-convergence is the analogue of r-convergence. This solves a question about the asymptotic optimality of Wald's sequent"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0510043","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"}