Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables
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 \textit{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.
asymptotically almost negative association
scientific article; zbMATH DE number 897187