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A STRONG LAW OF LARGE NUMBERS FOR AANA RANDOM VARIABLES IN A HILBERT SPACE AND ITS APPLICATION

  • Ko, Mi-Hwa (Department of Statistics, WonKwang University)
  • Received : 2010.01.12
  • Accepted : 2010.02.22
  • Published : 2010.03.25

Abstract

In this paper we introduce the concept of asymptotically almost negatively associated random variables in a Hilbert space and obtain the strong law of large numbers for a strictly stationary asymptotically almost negatively associated sequence of H-valued random variables with zero means and finite second moments. As an application we prove a strong law of large numbers for a linear process generated by asymptotically almost negatively random variables in a Hilbert space with this result.

Keywords

References

  1. Araujo, A. and Gine, E.(1980) The Central Limit Theorem for Real and Banach Valued Random Variables, John Wiley and Sons.
  2. Bosq, D. (2004) Berry-Essen inequality for linear processes in Hilbert spaces, Statist. Probab. Lett. 63 243-247
  3. Brockwell, P. and Davis, R.(1987) Time series, Theory and Method. Springer, Berlin
  4. Burton, R., Dabrowski, A.R. and Dehling, H.(1986) An invariance principle for weakly associated random vectors, Stochastic Processes Appl. 23 301-306 https://doi.org/10.1016/0304-4149(86)90043-8
  5. Chandra, T. K., Ghosal, S. (1996a). Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables, Acta. Math. Hungar. 32 327-336.
  6. Chandra, T. K., Ghosal, S. (1996b). The strong law of large numbers for weighted averages under dependence assumptions, J. Theor. Probab. 9 797-809. https://doi.org/10.1007/BF02214087
  7. Esary, J. Proschan, F. and Walkup, D. (1967). Association of random variables with applications, Ann. Math. Stat. 38 1466-1474. https://doi.org/10.1214/aoms/1177698701
  8. Joag-Dev, K. and Proschan, F. (1983) Negative association of random variables with applications, Ann. Statist. 11 286-295 https://doi.org/10.1214/aos/1176346079
  9. Kim, T.S., Ko, M.H. and Lee I.H. (2004). On the strong law for asymptotically almost negatively associated random variables, Rocky Mountain J. Math. 34 979-989. https://doi.org/10.1216/rmjm/1181069838
  10. Ko, M.H., Kim, T .S. and Han, K.H. (2009) A note on the almost sure convergence for dependent random variables in a Hilbert space, J. Them. Probab. 22 506-513 https://doi.org/10.1007/s10959-008-0144-z
  11. Ko, M.H. (2009) A central limit theorem for linear process in a Hilbert space under negative association, Korean Commun. Stat. 16 687-696 https://doi.org/10.5351/CKSS.2009.16.4.687
  12. Melevede, F., Peligrad, M. and Utev, S. (1997) Sharp conditions for the CLT of linear processes in a Hilbert space, J. Theor. Probab. 10 681-693 https://doi.org/10.1023/A:1022653728014
  13. Stout, W.F. (1995) Almost sure convergence, Academic, New York.