Abstract
Let \({\{X_n, n \geq1 \}}\) be a sequence of random variables and {b n , n ≥ 1} a nondecreasing sequence of positive constants. No assumptions are imposed on the joint distributions of the random variables. Some sufficient conditions are given under which \({\lim_{n\to \infty}\sum_{i=1}^n X_i/b_n=0}\) almost surely. Necessary conditions for the strong law of large numbers are also given.
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The research of T.-C. Hu has been supported by the Ministry of Science and Technology, R.O.C. (MOST 103-2118-M-007-002-MY2).
The research of S. H. Sung has been supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014R1A1A2058041).
The research of A.Volodin has been supported by the Natural Sciences and Engineering Research Council of Canada.
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Hu, TC., Sung, S.H. & Volodin, A. A note on the strong laws of large numbers for random variables. Acta Math. Hungar. 150, 412–422 (2016). https://doi.org/10.1007/s10474-016-0650-x
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DOI: https://doi.org/10.1007/s10474-016-0650-x