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A note on the almost sure convergence of sums of negatively dependent random variables. (English) Zbl 0925.60024

The results of this paper include the following: a) A necessary and sufficient condition for the validity of the strong law of large numbers when \(X\) is an identically distributed sequence of pairwise negative quadrant dependent r.v.’s (a result for r.v.’s with multidimensional indices is also given). b) If \(X\) is a sequence of negatively associated r.v.’s with finite second moments, the convergence of the series of the variances implies the almost sure convergence of \(\sum_{n=1}^\infty (X_n - EX_n)\).
A strong law of large numbers and the sufficiency part of the classical three series theorem are thus extended to this setting.

MSC:

60F15 Strong limit theorems
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[1] Birkel, T., A note on the strong law of large numbers for positively dependent random variables, Statist. Probab. Lett., 7, 17-20 (1989) · Zbl 0661.60048
[2] Esary, J.; Proschan, F.; Walkup, D., Association of random variables with applications, Ann. Math. Statist., 38, 1466-1474 (1967) · Zbl 0183.21502
[3] Etemadi, N., An elementary proof of the strong law of large numbers, Z. Wahrsch. Verw. Gebiete, 55, 119-122 (1981) · Zbl 0438.60027
[4] Etemadi, N., On the strong law of large numbers for nonnegative random variables, J. Multivariate Anal., 13, 187-193 (1983) · Zbl 0524.60033
[5] Joag-Dev, K.; Proschan, F., Negative association of random variables with applications, Ann. Statist., 11, 286-295 (1983) · Zbl 0508.62041
[6] Lehmann, E. L., Some concepts of dependence, Ann. Math. Statist., 37, 1137-1153 (1966) · Zbl 0146.40601
[7] Newman, C. M., Asymptotic independence and limit theorems for positively and negatively dependent random variables, (Tong, Y. L., Inequalities in Statistics and Probability (1984), Institute of Mathematical Statistics: Institute of Mathematical Statistics Hayward, CA), 127-140
[8] Petrov, W. W., Limit Theorems for Sums of Independent Random Variables (1987), Nauka: Nauka Moscow, [In Russian.] · Zbl 0621.60022
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