{"paper":{"title":"Toeplitz Lemma, Complete Convergence and Complete Moment Convergence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jiyanglin Li, Ze-Chun Hu","submitted_at":"2014-06-03T16:10:00Z","abstract_excerpt":"In this paper, we study the Toeplitz lemma, the Ces\\`{a}ro mean convergence theorem and the Kronecker lemma. At first, we study \"complete convergence\" versions of the Toeplitz lemma, the Ces\\`{a}ro mean convergence theorem and the Kronecker lemma. Two counterexamples show that they can fail in general and some sufficient conditions for \"complete convergence\" version of the Ces\\`{a}ro mean convergence theorem are given. Secondly  we introduce two classes of complete moment convergence, which are stronger versions of mean convergence and consider the Toeplitz lemma, the Ces\\`{a}ro mean convergen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.0768","kind":"arxiv","version":3},"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"}