Abstract
Three examples are provided which demonstrate that “convergence in probability” versions of the Toeplitz lemma, the Cesàro mean convergence theorem, and the Kronecker lemma can fail. “Mean convergence” versions of the Toeplitz lemma, Cesàro mean convergence theorem, and the Kronecker lemma are presented and a general mean convergence theorem for a normed sum of independent random variables is established. Some additional problems are posed.
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