## On complete convergence for arrays of rowwise dependent random variables.(English)Zbl 1120.60025

Summary: This paper establishes two results for complete convergence in the law of large numbers for arrays under $$\rho$$-mixing and $$\widetilde \rho$$-mixing association in rows. The authors extend several known results.

### MSC:

 60F15 Strong limit theorems
Full Text:

### References:

 [1] Ahmed, S.E.; Antonini, R.G.; Volodin, A., On complete convergence for weighted sums of arrays of Banach space valued random elements with application to moving average processes, Statist. probab. lett., 58, 185-194, (2002) · Zbl 1017.60013 [2] Baum, I.E.; Katz, M., Convergence rates in the law of large numbers, Trans. amer. math. soc., 120, 108-123, (1965) · Zbl 0142.14802 [3] Bozorgnia, A.; Patterson, R.F.; Taylor, R.L., On strong laws of large numbers for arrays of rowwise independent random elements, Internat. J. math. math. sci., 16, 587-592, (1993) · Zbl 0844.60005 [4] Chow, Y.S., Delayed sums and Borel summability of independent, identically distributed random variables, Bull. inst. math. acad. sinica, 207-220, (1973) · Zbl 0296.60014 [5] Erdös, P., On a theorem of hsu and robbins, Ann. math. statist., 20, 286-291, (1949) · Zbl 0033.29001 [6] Gut, A., Complete convergence and convergence rates for randomly indexed partial sums with an application to some first passage times, Acta math. hungar., 42, 225-232, (1983) · Zbl 0536.60036 [7] Gut, A., Complete convergence for arrays, Periodica math. hungar., 25, 51-75, (1992) · Zbl 0760.60029 [8] Hsu, P.L.; Robbins, H., Complete convergence and the law of large numbers, Proc. nat. acad. sci. USA, 33, 25-31, (1947) · Zbl 0030.20101 [9] Hu, T.C.; Volodin, A.I., Addendum to “A note on complete convergence for arrays”, Statist. probab. lett., 41, 209-211, (2000) [10] Hu, T.C.; Moricz, F.; Taylor, R.L., Strong law of large numbers for arrays of random variables, Acta math. acad. sci. hungar., 54, 153-162, (1989) · Zbl 0685.60032 [11] Hu, T.C.; Szynal, D.; Volodin, A.I., A note on complete convergence for arrays, Statist. probab. lett., 38, 27-31, (1998) · Zbl 0910.60017 [12] Hu, T.C.; Rosalsky, A.; Szynal, D.; Volodin, A.I., On complete convergence for arrays of rowwise independent random elements in Banach spaces, Stochastic anal. appl., 17, 963-992, (1999) · Zbl 0940.60032 [13] Katz, M.L., The probability in the tail of a distribution, Ann. math. statist., 34, 312-318, (1963) · Zbl 0209.49503 [14] Kuczmaszewska, A., On some condition for complete convergence for array, Statist. probab. lett., 66, 399-405, (2004) · Zbl 1074.60038 [15] Kuczmaszewska, A.; Szynal, D., On the hsu – robbins law of large numbers for subsequences, Bull. acad. Pol. math., 32, 729-735, (1988) · Zbl 0676.60027 [16] Kuczmaszewska, A.; Szynal, D., On complete convergence for partial sums of independent identically distributed random variables, Probab. math. statist., 11, 223-235, (1991) · Zbl 0748.60032 [17] Kuczmaszewska, A.; Szynal, D., On complete convergence in a Banach space, Internat. J. math. math. sci., 17, 1-14, (1994) · Zbl 0798.60006 [18] Peligrad, M.; Gut, A., Almost-sure results for a class of dependent random variables, J. theoret. probab., 12, 87-104, (1999) · Zbl 0928.60025 [19] Pruitt, W.E., Summability of independent random variables, J. math. mech., 15, 769-776, (1966) · Zbl 0158.36403 [20] Rohatgi, V.K., Convergence of weighted sums of independent random variables, Proc. Cambridge philos. soc., 69, 305-307, (1971) · Zbl 0209.20004 [21] Shao, Q.M., Maximal inequalities for partial sums of $$\varrho$$-mixing sequences, Ann. probab., 23, 948-965, (1995) · Zbl 0831.60028 [22] Shixin, G., Almost sure convergence for $$\widetilde{\varrho}$$-mixing random variable sequence, Statist. probab. lett., 67, 289-298, (2004) · Zbl 1043.60023 [23] Sung, S.H., Complete convergence for weighted sums of arrays of rowwise independent B-valued random variables, Stochastic anal. appl., 15, 255-267, (1997) · Zbl 0902.60011 [24] Sung, S.H.; Volodin, A.I.; Hu, T.C., On complete convergence for arrays, Statist. probab. lett., 71, 303-311, (2005) · Zbl 1087.60030 [25] Wang, X.; Rao, M.B.; Yang, X., Convergence rates on strong laws of large numbers for arrays of rowwise independent elements, Stochastic anal. appl., 11, 115-132, (1993) · Zbl 0764.60037 [26] Shanchao, Y., Some moment inequalities for partial sums of random variables and their applications, Chinese sci. bull., 43, 1823-1827, (1998) [27] Zhengyan, L., Chuanrong, L., 1996. Limit theory for mixing dependent random variables. Mathematics and its Applications, vol. 378. Kluwer Academic Publishers, Dordrecht; Science Press, New York. · Zbl 0889.60001
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.