# zbMATH — the first resource for mathematics

Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables. (English) Zbl 0853.60032
The classical strong law of large numbers (SLLN) has been extended to various weakly dependent random variables. However, another important SLLN, namely that of Marcinkiewicz and Zygmund (MZ), has not been extended before to dependent random variables. We obtain MZSLLN under several different dependence conditions. Unlike the usual proof of MZSLLN in the independent case which uses martingale methods, we use a suitable combination of the subsequence method and the method of maximal inequalities. It is observed that our proof works whenever a truncated version of the random variables admits a Kolmogorov type maximal inequality. We thus obtain MZSLLN for asymptotically almost negatively associated (AANA), $$\varphi$$-mixing and asymptotically quadrant subindependent (AQSI) random variables. Moreover, using the same technique, MZSLLNs for other dependent sequences may be obtained once suitable maximal inequalities are established. The AANA and AQSI conditions are introduced by the present authors and these are weakenings of negative association and asymptotically quadrant independence of T. Birkel [Stat. Probab. Lett. 7, No. 1, 17-20 (1988; Zbl 0661.60048)], respectively. A maximal inequality for AANA sequence is also established. The condition of identical distribution is relaxed to a great extent. Thus even for independent sequences, our results generalize the existing MZSLLN.

##### MSC:
 60F15 Strong limit theorems
Full Text:
##### References:
 [1] R. C. Bradley, W. Bryc and S. Janson, On dominations between measures of dependence. J. Mult. Anal., 3 (1987), 312-329. · Zbl 0627.60009 · doi:10.1016/0047-259X(87)90160-6 [2] T. Birkel, Laws of large numbers under dependence assumptions, Statist. Probab. Lett., 7 (1992), 17-20. · Zbl 0661.60048 · doi:10.1016/0167-7152(88)90080-6 [3] T.K. Chandra, Extensions of Rajchman’s strong law of large numbers, Sankhy?, Ser. A 53 (1991), 118-121. · Zbl 0749.60028 [4] T. K. Chandra and S. Ghosal, Some elementary strong laws of large numbers: a review, Tech. Report, Indian Statistical Institute (1993). · Zbl 0928.60020 [5] Y. S. Chow and H. Teicher, Probability Theory: Indepedence, Interchangeability, Martingales, Second Edition, Springer-Verlag (New York, 1988). [6] J. L. Doob, Stochastic Processes, John Wiley, (New York, 1953). [7] E. L. Lehmann, Some concepts of dependence, Ann. Math. Statist., 43 (1966), 1137-1153. · Zbl 0146.40601 · doi:10.1214/aoms/1177699260 [8] P. Matula, A note on the almost sure convergence of sums of negatively dependent random variables, Statist. Probab. Lett., 15, (1992), 209-213. · Zbl 0925.60024 · doi:10.1016/0167-7152(92)90191-7 [9] D. L. McLeish, A maximal inequality and dependent strong laws, Ann. Probab., 3 (1975), 829-839. · Zbl 0353.60035 · doi:10.1214/aop/1176996269 [10] M. Rosenblatt, Markov Processes, Structure and Asymptotic Behavior, Springer-Verlag (New York, 1971). · Zbl 0236.60002 [11] W. F. Stout, Almost Sure Convergence, Academic Press (New York, 1974). · Zbl 0321.60022
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.