Toeplitz lemma, complete convergence, and complete moment convergence

Jiyanglin Li and Ze-Chun Hu

Communications in Statistics - Theory and Methods, 2017, vol. 46, issue 4, 1731-1743

Abstract: In this paper, we study the Toeplitz lemma, the Cesàro mean convergence theorem, and the Kronecker lemma. At first, we study “complete convergence” versions of the Toeplitz lemma, the Cesàro 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 Cesàro mean convergence theorem are given. Second, we introduce two classes of complete moment convergence, which are stronger versions of mean convergence and consider the Toeplitz lemma, the Cesàro mean convergence theorem, and the Kronecker lemma under these two classes of complete moment convergence.

Date: 2017
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DOI: 10.1080/03610926.2015.1026996

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