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Uniform integrability in the Cesàro sense and the weak law of large numbers. (English) Zbl 0721.60024
After defining uniform integrability in the Cesàro sense, \(L^ 1\)- convergence of sample mean of pairwise independent random variables is established. Necessary and sufficient conditions for uniform integrability in the Cesàro sense are established. In the case of complete independence, it is shown by means of an example that uniform integrability in the Cesàro sense is not necessary to prove \(L^ 1\)- convergence of sample mean. It is also shown by an example that uniform integrability in the Cesàro mean is strictly weaker than the uniform integrability. Further, \(L'\)-convergence of sample mean is extended to the sequences of martingale-difference and \(\phi\)-mixing random variables.

MSC:
60F05 Central limit and other weak theorems
60G42 Martingales with discrete parameter
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