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Keywords:
negative dependence; pairwise negative dependence; Hilbert space; law of large numbers
Summary:
This paper introduces the notion of pairwise and coordinatewise negative dependence for random vectors in Hilbert spaces. Besides giving some classical inequalities, almost sure convergence and complete convergence theorems are established. Some limit theorems are extended to pairwise and coordinatewise negatively dependent random vectors taking values in Hilbert spaces. An illustrative example is also provided.
References:
[1] Block, H. W., Savits, T. H., Shaked, M.: Some concepts of negative dependence. Ann. Probab. 10 (1982), 765-772. DOI 10.1214/aop/1176993784 | MR 0659545 | Zbl 0501.62037
[2] Borcea, J., Brändén, P., Liggett, T. M.: Negative dependence and the geometry of polynomials. J. Am. Math. Soc. 22 (2009), 521-567. DOI 10.1090/S0894-0347-08-00618-8 | MR 2476782 | Zbl 1206.62096
[3] Burton, R. M., Dabrowski, A. R., Dehling, H.: An invariance principle for weakly associated random vectors. Ann. Probab. 9 (1981), 671-675. DOI 10.1214/aop/1176994374 | MR 0876052 | Zbl 0465.60009
[4] Chen, P., Sung, S. H.: Complete convergence and strong laws of large numbers for weighted sums of negatively orthant dependent random variables. Acta Math. Hung. 148 (2016), 83-95. DOI 10.1007/s10474-015-0559-9 | MR 3439284 | Zbl 1374.60040
[5] Csörgő, S., Tandori, K., Totik, V.: On the strong law of large numbers for pairwise independent random variables. Acta Math. Hung. 42 (1983), 319-330. DOI 10.1007/BF01956779 | MR 0722846 | Zbl 0534.60028
[6] Dabrowski, A. R., Dehling, H.: A Berry-Esséen theorem and a functional law of the iterated logarithm for weakly associated random vectors. Stochastic Processes Appl. 30 (1988), 277-289. DOI 10.1016/0304-4149(88)90089-0 | MR 0978359 | Zbl 0665.60027
[7] Ebrahimi, N., Ghosh, M.: Multivariate negative dependence. Commun. Stat., Theory Methods A10 (1981), 307-337. DOI 10.1080/03610928108828041 | MR 0612400 | Zbl 0506.62034
[8] Etemadi, N.: An elementary proof of the strong law of large numbers. Z. Wahrscheinlichkeitstheor. Verw. Geb. 55 (1981), 119-122. DOI 10.1007/BF01013465 | MR 0606010 | Zbl 0438.60027
[9] Hájek, J., Rényi, A.: Generalization of an inequality of Kolmogorov. Acta Math. Acad. Sci. Hung. 6 (1955), 281-283. DOI 10.1007/BF02024392 | MR 0076207 | Zbl 0067.10701
[10] Hien, N. T. T., Thanh, L. V.: On the weak laws of large numbers for sums of negatively associated random vectors in Hilbert spaces. Stat. Probab. Lett. 107 (2015), 236-245. DOI 10.1016/j.spl.2015.08.030 | MR 3412782 | Zbl 1329.60029
[11] Hu, T.-C., Sung, S. H., Volodin, A.: A note on the strong laws of large numbers for random variables. Acta Math. Hung. 150 (2016), 412-422. DOI 10.1007/s10474-016-0650-x | MR 3568100 | Zbl 06842519
[12] Huan, N. V., Quang, N. V., Thuan, N. T.: Baum-Katz type theorems for coordinatewise negatively associated random vectors in Hilbert spaces. Acta Math. Hung. 144 (2014), 132-149. DOI 10.1007/s10747-014-0424-2 | MR 3267175 | Zbl 1349.60046
[13] Ko, M.-H.: Hájek-Rényi inequality for $m$-asymptotically almost negatively associated random vectors in Hilbert space and applications. J. Inequal. Appl. (2018), Paper No. 80, 9 pages. DOI 10.1186/s13660-018-1671-5 | MR 3797137
[14] Ko, M.-H., Kim, T.-S., Han, K.-H.: A note on the almost sure convergence for dependent random variables in a Hilbert space. J. Theor. Probab. 22 (2009), 506-513. DOI 10.1007/s10959-008-0144-z | MR 2501332 | Zbl 1166.60021
[15] Lehmann, E. L.: Some concepts of dependence. Ann. Math. Stat. 37 (1966), 1137-1153. DOI 10.1214/aoms/1177699260 | MR 0202228 | Zbl 0146.40601
[16] 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 (2006), 281-305. MR 2230590 | Zbl 1102.60026
[17] Li, R., Yang, W.: Strong convergence of pairwise NQD random sequences. J. Math. Anal. Appl. 344 (2008), 741-747. DOI 10.1016/j.jmaa.2008.02.053 | MR 2426304 | Zbl 1141.60012
[18] Matuł{a}, P.: A note on the almost sure convergence of sums of negatively dependent random variables. Stat. Probab. Lett. 15 (1992), 209-213. DOI 10.1016/0167-7152(92)90191-7 | MR 1190256 | Zbl 0925.60024
[19] Miao, Y.: Hájek-Rényi inequality for dependent random variables in Hilbert space and applications. Rev. Unión Mat. Argent. 53 (2012), 101-112. MR 2987160 | Zbl 1255.60035
[20] Móricz, F.: Strong limit theorems for blockwise $m$-dependent and blockwise quasi-orthogonal sequences of random variables. Proc. Am. Math. Soc. 101 (1987), 709-715. DOI 10.2307/2046676 | MR 0911038 | Zbl 0632.60025
[21] Móricz, F., Su, K.-L., Taylor, R. L.: Strong laws of large numbers for arrays of orthogonal random elements in Banach spaces. Acta Math. Hung. 65 (1994), 1-16. DOI 10.1007/BF01874465 | MR 1275656 | Zbl 0806.60002
[22] Móricz, F., Taylor, R. L.: Strong laws of large numbers for arrays of orthogonal random variables. Math. Nachr. 141 (1989), 145-152. DOI 10.1002/mana.19891410116 | MR 1014423 | Zbl 0674.60006
[23] Patterson, R. F., Taylor, R. L.: Strong laws of large numbers for negatively dependent random elements. Nonlinear Anal., Theory Methods Appl. 30 (1997), 4229-4235. DOI 10.1016/S0362-546X(97)00279-4 | MR 1603567 | Zbl 0901.60016
[24] Pemantle, R.: Towards a theory of negative dependence. J. Math. Phys. 41 (2000), 1371-1390. DOI 10.1063/1.533200 | MR 1757964 | Zbl 1052.62518
[25] Rosalsky, A., Thanh, L. V.: On the strong law of large numbers for sequences of blockwise independent and blockwise $p$-orthogonal random elements in Rademacher type $p$ Banach spaces. Probab. Math. Stat. 27 (2007), 205-222. MR 2445993 | Zbl 1148.60018
[26] Rosalsky, A., Thanh, L. V.: Some strong laws of large numbers for blockwise martingale difference sequences in martingale type $p$ Banach spaces. Acta Math. Sin., Engl. Ser. 28 (2012), 1385-1400. DOI 10.1007/s10114-012-0378-7 | MR 2928485 | Zbl 1271.60012
[27] Thanh, L. V.: On the almost sure convergence for dependent random vectors in Hilbert spaces. Acta Math. Hung. 139 (2013), 276-285. DOI 10.1007/s10474-012-0275-7 | MR 3044151 | Zbl 1299.60042
[28] Wu, Y., Rosalsky, A.: Strong convergence for $m$-pairwise negatively quadrant dependent random variables. Glas. Mat., III. Ser. 50 (2015), 245-259. DOI 10.3336/gm.50.1.15 | MR 3361275 | Zbl 1323.60053
[29] Zhang, L.-X.: Strassen's law of the iterated logarithm for negatively associated random vectors. Stochastic Processes Appl. 95 (2001), 311-328. DOI 10.1016/S0304-4149(01)00107-7 | MR 1854030 | Zbl 1059.60042
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