Toeplitz lemma, complete convergence, and complete moment convergence

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.

KEYWORDS:

• Toeplitz lemma
• Cesàro mean convergence theorem
• Kronecker lemma
• complete convergence
• complete moment convergence

Mathematics Subject Classification:

• 60F05
• 60F15
• 60F25
• 40A05

Acknowledgments

We acknowledge the helpful suggestions and comments of three anonymous referees, which improved the presentation of this paper.

Funding

We are grateful to the support of NNSFC (Grant No. 11371191) and Jiangsu Province Basic Research Program (Natural Science Foundation) (Grant No. BK2012720).