Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON A FUNCTIONAL CENTRAL LIMIT THEOREM FOR THE LINEAR PROCESS GENERATED BY ASSOCIATED RANDOM VARIABLES IN A HILBERT SPACE

  • Ko, Mi-Hwa (DEPARTMENT OF MATHEMATICS WONKWANG UNIVERSITY) ;
  • Kim, Tae-Sung (DEPARTMENT OF MATHEMATICS WONKWANG UNIVERSITY)
  • Published : 2008.01.31
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Abstract

Let {${\xi}_k,\;k\;{\in}\;{\mathbb{Z}}$} be a strictly stationary associated sequence of H-valued random variables with $E{\xi}_k\;=\;0$ and $E{\parallel}{\xi}_k{\parallel}^2\;<\;{\infty}$ and {$a_k,\;k\;{\in}\;{\mathbb{Z}}$} a sequence of linear operators such that ${\sum}_{j=-{\infty}}^{\infty}\;{\parallel}a_j{\parallel}_{L(H)}\;<\;{\infty}$. For a linear process $X_k\;=\;{\sum}_{j=-{\infty}}^{\infty}\;a_j{\xi}_{k-j}$ we derive that {$X_k} fulfills the functional central limit theorem.

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References

  1. R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing : Probability models Holt, Rinehart and Winston, New York, 1975
  2. P. Billingsley, Convergence of Probability, John Wiley, New York, 1968
  3. D. Bosq, Linear Processes in Function Spaces, Lectures Notes in Statistics, 149, Springer, Berlin, 2000
  4. D. Bosq, Berry-Esseen inequality for linear processes in Hilbert spaces, Statist. Probab. Letters 63 (2003), 243-247 https://doi.org/10.1016/S0167-7152(03)00088-9
  5. P. Brockwell and R. Davis, Time Series, Theory and Method. Springer, Berlin, 1987
  6. R. A. R. Burton and H. Dehling, An invariance principle for weakly associated random vectors, Stochastic Processes Appl. 23 (1986), 301-306 https://doi.org/10.1016/0304-4149(86)90043-8
  7. J. T. Cox and G. Grimmett, Central limit theorems for associated random variables and the percolation model, Ann. Probab. 12 (1984), 514-528 https://doi.org/10.1214/aop/1176993303
  8. N. A. Denisevskii and Y. A. Dorogovtsev, On the law of large numbers for a linear process in Banach space, Soviet Math. Dokl. 36 (1988), no. 1, 47-50
  9. T. S. Kim, M. H. Ko, and S. M. Chung, A central limit theorem for the stationary multivariate linear processes generated by associated random vectors, Commun. Korean Math. Soc. 17 (2002), no. 1, 95-102 https://doi.org/10.4134/CKMS.2002.17.1.095
  10. T. S. Kim and M. H. Ko, On a functional central limit theorem for stationary linear processes generated by associated processes, Bull. Korean Math. Soc. 40 (2003), no. 4, 715-722 https://doi.org/10.4134/BKMS.2003.40.4.715
  11. C. M. Newman, Normal fluctuations and the FKG inequalities, Comm. Math. Phys. 74 (1980), 119-128 https://doi.org/10.1007/BF01197754
  12. C. M. Newman and A. L. Wright, An invariance principle for certain dependent sequences, Ann. Probab. 9 (1981), 671-675 https://doi.org/10.1214/aop/1176994374

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  1. Precise asymptotics for the linear processes generated by associated random variables in Hilbert spaces vol.64, pp.6, 2012, https://doi.org/10.1016/j.camwa.2012.03.046