On the negative dependence in Hilbert spaces with applications

Nguyen Thi Thanh Hien; Le Van Thanh; Vo Thi Hong Van

Applications of Mathematics (2019)

  • Volume: 64, Issue: 1, page 45-59
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Hien, Nguyen Thi Thanh, Thanh, Le Van, and Van, Vo Thi Hong. "On the negative dependence in Hilbert spaces with applications." Applications of Mathematics 64.1 (2019): 45-59. <http://eudml.org/doc/294534>.

@article{Hien2019,
abstract = {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.},
author = {Hien, Nguyen Thi Thanh, Thanh, Le Van, Van, Vo Thi Hong},
journal = {Applications of Mathematics},
keywords = {negative dependence; pairwise negative dependence; Hilbert space; law of large numbers},
language = {eng},
number = {1},
pages = {45-59},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the negative dependence in Hilbert spaces with applications},
url = {http://eudml.org/doc/294534},
volume = {64},
year = {2019},
}

TY - JOUR
AU - Hien, Nguyen Thi Thanh
AU - Thanh, Le Van
AU - Van, Vo Thi Hong
TI - On the negative dependence in Hilbert spaces with applications
JO - Applications of Mathematics
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 1
SP - 45
EP - 59
AB - 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.
LA - eng
KW - negative dependence; pairwise negative dependence; Hilbert space; law of large numbers
UR - http://eudml.org/doc/294534
ER -

References

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  1. Block, H. W., Savits, T. H., Shaked, M., 10.1214/aop/1176993784, Ann. Probab. 10 (1982), 765-772. (1982) Zbl0501.62037MR0659545DOI10.1214/aop/1176993784
  2. Borcea, J., Brändén, P., Liggett, T. M., 10.1090/S0894-0347-08-00618-8, J. Am. Math. Soc. 22 (2009), 521-567. (2009) Zbl1206.62096MR2476782DOI10.1090/S0894-0347-08-00618-8
  3. Burton, R. M., Dabrowski, A. R., Dehling, H., 10.1214/aop/1176994374, Ann. Probab. 9 (1981), 671-675. (1981) Zbl0465.60009MR0876052DOI10.1214/aop/1176994374
  4. Chen, P., Sung, S. H., 10.1007/s10474-015-0559-9, Acta Math. Hung. 148 (2016), 83-95. (2016) Zbl1374.60040MR3439284DOI10.1007/s10474-015-0559-9
  5. Csörgő, S., Tandori, K., Totik, V., 10.1007/BF01956779, Acta Math. Hung. 42 (1983), 319-330. (1983) Zbl0534.60028MR0722846DOI10.1007/BF01956779
  6. Dabrowski, A. R., Dehling, H., 10.1016/0304-4149(88)90089-0, Stochastic Processes Appl. 30 (1988), 277-289. (1988) Zbl0665.60027MR0978359DOI10.1016/0304-4149(88)90089-0
  7. Ebrahimi, N., Ghosh, M., 10.1080/03610928108828041, Commun. Stat., Theory Methods A10 (1981), 307-337. (1981) Zbl0506.62034MR0612400DOI10.1080/03610928108828041
  8. Etemadi, N., 10.1007/BF01013465, Z. Wahrscheinlichkeitstheor. Verw. Geb. 55 (1981), 119-122. (1981) Zbl0438.60027MR0606010DOI10.1007/BF01013465
  9. Hájek, J., Rényi, A., 10.1007/BF02024392, Acta Math. Acad. Sci. Hung. 6 (1955), 281-283. (1955) Zbl0067.10701MR0076207DOI10.1007/BF02024392
  10. Hien, N. T. T., Thanh, L. V., 10.1016/j.spl.2015.08.030, Stat. Probab. Lett. 107 (2015), 236-245. (2015) Zbl1329.60029MR3412782DOI10.1016/j.spl.2015.08.030
  11. Hu, T.-C., Sung, S. H., Volodin, A., 10.1007/s10474-016-0650-x, Acta Math. Hung. 150 (2016), 412-422. (2016) Zbl06842519MR3568100DOI10.1007/s10474-016-0650-x
  12. Huan, N. V., Quang, N. V., Thuan, N. T., 10.1007/s10747-014-0424-2, Acta Math. Hung. 144 (2014), 132-149. (2014) Zbl1349.60046MR3267175DOI10.1007/s10747-014-0424-2
  13. Ko, M.-H., 10.1186/s13660-018-1671-5, J. Inequal. Appl. (2018), Paper No. 80, 9 pages. (2018) MR3797137DOI10.1186/s13660-018-1671-5
  14. Ko, M.-H., Kim, T.-S., Han, K.-H., 10.1007/s10959-008-0144-z, J. Theor. Probab. 22 (2009), 506-513. (2009) Zbl1166.60021MR2501332DOI10.1007/s10959-008-0144-z
  15. Lehmann, E. L., 10.1214/aoms/1177699260, Ann. Math. Stat. 37 (1966), 1137-1153. (1966) Zbl0146.40601MR0202228DOI10.1214/aoms/1177699260
  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. (2006) Zbl1102.60026MR2230590
  17. Li, R., Yang, W., 10.1016/j.jmaa.2008.02.053, J. Math. Anal. Appl. 344 (2008), 741-747. (2008) Zbl1141.60012MR2426304DOI10.1016/j.jmaa.2008.02.053
  18. Matuł{a}, P., 10.1016/0167-7152(92)90191-7, Stat. Probab. Lett. 15 (1992), 209-213. (1992) Zbl0925.60024MR1190256DOI10.1016/0167-7152(92)90191-7
  19. Miao, Y., Hájek-Rényi inequality for dependent random variables in Hilbert space and applications, Rev. Unión Mat. Argent. 53 (2012), 101-112. (2012) Zbl1255.60035MR2987160
  20. Móricz, F., 10.2307/2046676, Proc. Am. Math. Soc. 101 (1987), 709-715. (1987) Zbl0632.60025MR0911038DOI10.2307/2046676
  21. Móricz, F., Su, K.-L., Taylor, R. L., 10.1007/BF01874465, Acta Math. Hung. 65 (1994), 1-16. (1994) Zbl0806.60002MR1275656DOI10.1007/BF01874465
  22. Móricz, F., Taylor, R. L., 10.1002/mana.19891410116, Math. Nachr. 141 (1989), 145-152. (1989) Zbl0674.60006MR1014423DOI10.1002/mana.19891410116
  23. Patterson, R. F., Taylor, R. L., 10.1016/S0362-546X(97)00279-4, Nonlinear Anal., Theory Methods Appl. 30 (1997), 4229-4235. (1997) Zbl0901.60016MR1603567DOI10.1016/S0362-546X(97)00279-4
  24. Pemantle, R., 10.1063/1.533200, J. Math. Phys. 41 (2000), 1371-1390. (2000) Zbl1052.62518MR1757964DOI10.1063/1.533200
  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. (2007) Zbl1148.60018MR2445993
  26. Rosalsky, A., Thanh, L. V., 10.1007/s10114-012-0378-7, Acta Math. Sin., Engl. Ser. 28 (2012), 1385-1400. (2012) Zbl1271.60012MR2928485DOI10.1007/s10114-012-0378-7
  27. Thanh, L. V., 10.1007/s10474-012-0275-7, Acta Math. Hung. 139 (2013), 276-285. (2013) Zbl1299.60042MR3044151DOI10.1007/s10474-012-0275-7
  28. Wu, Y., Rosalsky, A., 10.3336/gm.50.1.15, Glas. Mat., III. Ser. 50 (2015), 245-259. (2015) Zbl1323.60053MR3361275DOI10.3336/gm.50.1.15
  29. Zhang, L.-X., 10.1016/S0304-4149(01)00107-7, Stochastic Processes Appl. 95 (2001), 311-328. (2001) Zbl1059.60042MR1854030DOI10.1016/S0304-4149(01)00107-7

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