# zbMATH — the first resource for mathematics

Complete convergence theorems for normed row sums from an array of rowwise pairwise negative quadrant dependent random variables with application to the dependent bootstrap. (English) Zbl 1363.60038
Summary: Let $$\{X_{n,j}, 1\leq j\leq m(n), n\geq 1\}$$ be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let $$0<b_n\rightarrow \infty$$. Conditions are given for $$\sum \nolimits_{j=1}^{m(n)}X_{n,j}/b_n\rightarrow 0$$ completely and for $$\max \nolimits_{1\leq k\leq m(n)}\Bigl | \sum \nolimits_{j=1}^kX_{n,j}\Big | /b_n\rightarrow 0$$ completely. As an application of these results, we obtain a complete convergence theorem for the row sums $$\sum \nolimits_{j=1}^{m(n)}X_{n,j}^*$$ of the dependent bootstrap samples $$\{\{X_{n,j}^*, 1\leq j\leq m(n)\}, n\geq 1\}$$ arising from a sequence of i.i.d. random variables $$\{X_n, n\geq 1\}$$.
##### MSC:
 60F15 Strong limit theorems 62F40 Bootstrap, jackknife and other resampling methods 62G09 Nonparametric statistical resampling methods
Full Text:
##### References:
 [1] Bozorgnia, A.; Patterson, R. F.; Taylor, R. L., On strong laws of large numbers for arrays of rowwise independent random elements, Int. J. Math. Math. Sci., 16, 587-591, (1993) · Zbl 0844.60005 [2] Bozorgnia, A.; Patterson, R. F.; Taylor, R. L.; Lakshmikantham, V. (ed.), Limit theorems for dependent random variables, No. I-IV, 1639-1650, (1996), Berlin · Zbl 0845.60010 [3] Bozorgnia, A.; Patterson, R. F.; Taylor, R. L., Chung type strong laws for arrays of random elements and bootstrapping, Stochastic Anal. Appl., 15, 651-669, (1997) · Zbl 0899.60028 [4] Chung, K.-L., Note on some strong laws of large numbers, Am. J. Math., 69, 189-192, (1947) · Zbl 0034.07103 [5] Efron, B., Bootstrap methods: another look at the jackknife, Ann. Stat., 7, 1-26, (1979) · Zbl 0406.62024 [6] Erdős, P., On a theorem of hsu and robbins, Ann. Math. Stat., 20, 286-291, (1949) · Zbl 0033.29001 [7] Gan, S.; Chen, P., Some limit theorems for sequences of pairwise NQD random variables, Acta Math. Sci., Ser. B, Engl. Ed., 28, 269-281, (2008) · Zbl 1174.60330 [8] Hsu, P. L.; Robbins, H., Complete convergence and the law of large numbers, Proc. Natl. Acad. Sci. USA, 33, 25-31, (1947) · Zbl 0030.20101 [9] Hu, T. -C.; Moricz, F.; Taylor, R. L., Strong laws of large numbers for arrays of rowwise independent random variables, Acta Math. Hung., 54, 153-162, (1989) · Zbl 0685.60032 [10] Hu, T. -C.; Cabrera, M. O.; Volodin, A., Almost sure lim sup behavior of dependent bootstrap means, Stochastic Anal. Appl., 24, 939-952, (2006) · Zbl 1115.62044 [11] Hu, T. -C.; Taylor, R. L., On the strong law for arrays and for the bootstrap Mean and variance, Int. J. Math. Math. Sci., 20, 375-382, (1997) · Zbl 0883.60024 [12] Lehmann, E. L., Some concepts of dependence, Ann. Math. Stat., 37, 1137-1153, (1966) · Zbl 0146.40601 [13] Li, D.; Rosalsky, A.; Volodin, A. I., On the strong law of large numbers for sequences of pairwise negative quadrant dependent random variables, Bull. Inst. Math., Acad. Sin. (N. S.), 1, 281-305, (2006) · Zbl 1102.60026 [14] Matuła, P., A note on the almost sure convergence of sums of negatively dependent random variables, Stat. Probab. Lett., 15, 209-213, (1992) · Zbl 0925.60024 [15] Patterson, R. F.; Smith, W. D.; Taylor, R. L.; Bozorgnia, A., Limit theorems for negatively dependent random variables, Nonlinear Anal., Theory Methods Appl., 47, 1283-1295, (2001) · Zbl 1042.60503 [16] Patterson, R. F.; Taylor, R. L., Strong laws of large numbers for negatively dependent random elements, Nonlinear Anal., Theory Methods Appl., 30, 4229-4235, (1997) · Zbl 0901.60016 [17] Pemantle, R., Towards a theory of negative dependence, J. Math. Phys., 41, 1371-1390, (2000) · Zbl 1052.62518 [18] Smith, W. D.; Taylor, R. L., Consistency of dependent bootstrap estimators, Am. J. Math. Manage. Sci., 21, 359-382, (2001) [19] Smith, W. D.; Taylor, R. L., Dependent bootstrap confidence intervals, No. 37, 91-107, (2001), Beachwood [20] 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 [21] Volodin, A.; Cabrera, M. O.; Hu, T. C., Convergence rate of the dependent boot-strapped means, Theory Probab. Appl., 50, 337-346, (2006) · Zbl 1158.62036 [22] Wu, Q., Convergence properties of pairwise NQD random sequences, Acta Math. Sin., 45, 617-624, (2002) · Zbl 1008.60039 [23] Wu, Y.; Wang, D., Convergence properties for arrays of rowwise pairwise negatively quadrant dependent random variables, Appl. Math., Praha, 57, 463-476, (2012) · Zbl 1265.60067 [24] Wu, Y. -F.; Zhu, D. -J., Convergence properties of partial sums for arrays of rowwise negatively orthant dependent random variables, J. Korean Stat. Soc., 39, 189-197, (2010) · Zbl 1294.60056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.