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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References
Baum L.E., Katz M.: Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120, 108–123 (1965)
Chobanyan S., Levental S., Mandrekar V.: Prokhorov blocks and strong law of large numbers under rearrangements. J. Theoret. Probab. 17, 647–672 (2004)
Fazekas I., Klesov O.: A general approach to the strong law of large numbers. Theory Probab. Appl. 45, 436–449 (2001)
Hsu P.L., Robbins H.: Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. USA 33, 25–31 (1947)
Hu S., Hu M.: A general approach rate to the strong law of large numbers, Statist. Probab. Lett., 76, 843–851 (2006)
Klesov O., Rosalsky A., Volodin A.: On the almost sure growth rate of sums of lower negatively dependent nonnegative random variables. Statist. Probab. Lett. 71, 193–202 (2005)
M. Loéve, Probability Theory, vol. 1, Springer-Verlag (New York, 1997).
W. F. Stout, Almost Sure Convergence, Academic Press (New York, 1974).
Sung S.H., Hu T.-C., Volodin A.: A note on the growth rate in the Fazekas–Klesov general law of large numbers and on the weak law of large numbers for tail series. Publ. Math. Debrecen 73, 1–10 (2008)
Thanh L.V.: On the strong law of large numbers for d-dimensional arrays of random variables. Elect. Comm. Probab. 12, 434–441 (2007)
Tómács T.: A general method to obtain the rate of convergence in the strong law of large numbers. Ann. Math. Inform. 34, 97–102 (2007)
Tómács T., Líbor Z.: A Hájek–Rènyi type inequality and its applications. Ann. Math. Inform. 33, 141–149 (2006)
S. Yang, C. Su and K. Yu, A general method to the strong law of large numbers and its applications, Statist. Probab. Lett., 78 (2008), 794–803.
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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