Abstract
Complete convergence theorems are obtained for an array of rowwise extended negatively dependent random variables. Special cases of the main results are presented as corollaries. Illustrative examples are also presented.
Similar content being viewed by others
References
Cai, G.-h. (2011). Complete convergence of rowwise \(\overset {\sim }{ \rho }\)-mixing sequences of random variables. J. Appl. Funct. Anal. 6, 356–364.
Chow, Y. S. (1966). Some convergence theorems for independent random variables. Ann. Math. Stat. 37, 1482–1493.
Erdös, P. (1949). On a theorem of Hsu and Robbins. Ann. Math. Stat. 20, 286–291.
Ghosal, S., and Chandra, T. K. (1998). Complete convergence of martingale arrays. J. Theoretical Probab. 11, 621–631.
Gut, A. (1985). On complete convergence in the law of large numbers for subsequences. Ann. Probab. 13, 1286–1291.
Gut, A. (1992). Complete convergence for arrays. Period. Math. Hung. 25, 51–75.
Hsu, P. L., and Robbins, H. (1947). Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A. 33, 25–31.
Hu, T.-C., Rosalsky, A., and Volodin, A. (2012). A complete convergence theorem for row sums from arrays of rowwise independent random elements in Rademacher type p Banach spaces. Stoch. Anal. Appl. 30, 343–353.
Hu, T.-C., Szynal, D., and Volodin, A. I. (1998). A note on complete convergence for arrays. Stat. Probab. Lett. 38, 27–31.
Hu, T.-C., and Volodin, A. (2000). Addendum to “A note on complete convergence for arrays”, [Statist Probab. Lett. 38 (1998), no.1, 27-31]. Stat. Probab. Lett. 47, 209–211.
Huang, H., Wang, D., Wu, Q., and Peng, J. (2013). Complete convergence of pairwise NQD random sequences. Chinese J. Appl. Probab. Stat. 29, 275–286.
Joag-Dev, K and Proschan, F. (1983). Negative association of random variables, with applications. Ann. Stat. 11, 286–295.
Ko, M.-H. (2013). Complete convergence for arrays of row-wise PNQD random variables. Stochastics 85, 172–180.
Kuczmaszewska, A. (2010). On complete convergence in Marcinkiewicz-Zygmund type SLLN for negatively associated random variables. Acta Math. Hung. 128, 116–130.
Lehmann, E. (1966). Some concepts of dependence. Ann. Math. Stat. 37, 1137–1153.
Li, D., Rao, M. B., Jiang, T., and Wang, X. (1995). Complete convergence and almost sure convergence of weighted sums of random variables. J. Theor. Probab. 8, 49–76.
Liu, L. (2009). Precise large deviations for dependent random variables with heavy tails. Stat. Probab. Lett. 79, 1290–1298.
Liu, L. (2010). Necessary and sufficient conditions for moderate deviations of dependent random variables with heavy tails. Sci. China Math. 53, 1421–1434.
Qiu, D., Chen, P., Antonini, R. G., and Volodin, A. (2013). On the complete convergence for arrays of rowwise extended negatively dependent random variables. J. Korean Math. Soc. 50, 379–392.
Rohatgi, V. K. (1971). Convergence of weighted sums of independent random variables. Proc. Camb. Philos. Soc. 69, 305–307.
Shao, Q.-M. (1993). Complete convergence for α-mixing sequences. Stat. Probab. Lett. 16, 279–287.
Shao, Q.-M. (1995). Maximal inequalities for partial sums of ρ-mixing sequences. Ann. Probab. 23, 948–965.
Sung, S. H. (2012a). Complete convergence for weighted sums of negatively dependent random variables. Stat. Papers 53, 73–82.
Sung, S. H. (2012b). On complete convergnece for arrays of dependent random variables. Comm. Stat. Theory Methods 41, 1663–1674.
Sung, S. H. (2013). On the strong convergence for weighted sums of ρ ∗-mixing random variables. Stat. Papers 54, 773–781.
Sung, S. H., Volodin, A. I., and Hu, T.-C. (2005). More on complete convergence for arrays. Stat. Probab. Lett. 71, 303–311.
Wang, X. J., and Hu, S. H. (2014). Complete convergence and complete moment convergence for martingale difference sequence. Acta Math. Sinica (English Ser.) 30, 119–132.
Wang, X., Hu, S., Yang, W., and Li X. (2010). Exponential inequalities and complete convergence for a LNQD sequence. J. Korean Stat. Soc. 39, 555–564.
Wang, X., Li, X., Yang, W., and Hu, S. (2012). On complete convergence for arrays of rowwise weakly dependent random variables. Appl. Math. Lett. 25, 1916–1920.
Wang, X., Hu, T.-C., Volodin, A., and Hu, S. (2013a). Complete convergence for weighted sums and arrays of rowwise extended negatively dependent random variables. Comm. Stat. Theory Methods 42, 2391–2401.
Wang, X., Wang, S., Hu, S., Ling, J., and Wei, Y. (2013b). On complete convergence of weighted sums for arrays of rowwise extended negatively dependent random variables. Stochastics 85, 1060–1072.
Wu, Y. (2014). Strong convergence for weighted sums of arrays of rowwise pairwise NQD random variables. Collect. Math. 65, 119–130.
Wu, Y., Guan, M. (2012). Convergence properties of the partial sums for sequences of END random variables. J. Korean Math. Soc. 49, 1097–1110.
Wu, Y., Rosalsky, A., and Volodin, A. (2013). Some mean convergence and complete convergence theorems for sequences of m-linearly negative quadrant dependent random variables. Appl. Math. 58, 511–529.
Yu, K. F. (1990) Complete convergence of weighted sums of martingale differences. J. Theor. Probab. 3, 339–347.
Zhou, X.-c., and Lin, J.-g. (2013). On complete convergence for strong mixing sequences. Stochastics 85, 262–271.
Zhou, X.-c., Lin, J.-g., Wang, X.-j., and Hu, S.-h. (2011). On complete convergence for arrays of rowwise strong mixing random variables. Commun. Math. Res. 27, 234–242.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hu, TC., Wang, KL. & Rosalsky, A. Complete Convergence Theorems for Extended Negatively Dependent Random Variables. Sankhya A 77, 1–29 (2015). https://doi.org/10.1007/s13171-014-0058-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13171-014-0058-z
Keywords and phrases.
- Complete convergence
- Extended negatively dependent random variables
- Array of random variables
- Stochastic domination
- Weighted sums.