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On the almost sure growth rate of sums of lower negatively dependent nonnegative random variables. (English) Zbl 1070.60030
Summary: For a sequence of lower negatively dependent nonnegative random variables $$\{X_n$$, $$n\geq 1\}$$, conditions are provided under which $$\lim_{n\to\infty} \sum_{j=1}^n X_j/b_n= \infty$$ almost surely where $$\{b_n$$, $$n\geq 1\}$$ is a nondecreasing sequence of positive constants. The results are new even when they are specialized to the case of nonnegative independent and identically distributed summands and $$b_n= n^r$$, $$n\geq 1$$, where $$r>0$$.

##### MSC:
 60F15 Strong limit theorems
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##### References:
 [1] Adler, A.; Rosalsky, A.; Taylor, R.L., Some strong laws of large numbers for sums of random elements, Bull. inst. math. acad. sinica, 20, 335-357, (1992) · Zbl 0780.60009 [2] Amini, M.; Bozorgnia, A., Negatively dependent bounded random variable probability inequalities and the strong law of large numbers, J. appl. math. stochastic anal., 13, 261-267, (2000) · Zbl 1070.60028 [3] Chatterji, S.D., A general strong law, Invent. math., 9, 235-245, (1969/1970) · Zbl 0193.09301 [4] Derman, C.; Robbins, H., The strong law of large numbers when the first moment does not exist, Proc. nat. acad. sci. U.S.A., 41, 586-587, (1955) · Zbl 0064.38202 [5] Erickson, K.B., Recurrence sets of normed random walk in $$R^d$$, Ann. probab., 4, 802-828, (1976) · Zbl 0362.60074 [6] Gut, A.; Klesov, O.; Steinebach, J., Equivalences in strong limit theorems for renewal counting processes, Statist. probab. lett., 35, 381-394, (1997) · Zbl 0885.60026 [7] Kim, T.-S.; Baek, J.I., The strong laws of large numbers for weighted sums of pairwise quadrant dependent random variables, J. Korean math. soc., 36, 37-49, (1999) · Zbl 0928.60021 [8] Kim, T.-S.; Kim, H.-C., On the law of large numbers for weighted sums of pairwise negative quadrant dependent random variables, Bull. Korean math. soc., 38, 55-63, (2001) · Zbl 0980.60033 [9] Martikainen, A.I.; Petrov, V.V., On a theorem of Feller, Teor. veroyatnost. i primenen., 25, 194-197, (1980), (in Russian) (English translation: Theory Probab. Appl. 25, 191-193) · Zbl 0419.60025 [10] Matuła, P., A note on the almost sure convergence of sums of negatively dependent random variables, Statist. probab. lett., 15, 209-213, (1992) · Zbl 0925.60024 [11] Rosalsky, A., On the almost certain limiting behavior of normed sums of identically distributed positive random variables, Statist. probab. lett., 16, 65-70, (1993) · Zbl 0765.60020 [12] Sawyer, S., Maximal inequalities of weak type, Ann. of math. (2), 84, 157-174, (1966) · Zbl 0186.20503 [13] Taylor, R.L.; Patterson, R.F.; Bozorgnia, A., A strong law of large numbers for arrays of rowwise negatively dependent random variables, Stochastic anal. appl., 20, 643-656, (2002) · Zbl 1003.60032
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