Toeplitz Lemma, Complete Convergence and Complete Moment Convergence

TL;DR

This paper investigates the complete convergence versions of classical lemmas in analysis, provides counterexamples, and introduces stronger classes of complete moment convergence to extend these results.

Contribution

It introduces and analyzes complete convergence versions of the Toeplitz, Cesàro, and Kronecker lemmas, including counterexamples and new classes of complete moment convergence.

Findings

Counterexamples show failure of complete convergence versions in general

Sufficient conditions for Cesàro mean convergence under complete convergence

Introduction of stronger classes of complete moment convergence

Abstract

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 convergence theorem, and the Kronecker lemma under these two classes of complete moment convergence.