A Central Limit Theorem for the Linear Process in a Hilbert Space under Negative Association

- Journal title : Communications for Statistical Applications and Methods
- Volume 16, Issue 4, 2009, pp.687-696
- Publisher : The Korean Statistical Society
- DOI : 10.5351/CKSS.2009.16.4.687

Title & Authors

A Central Limit Theorem for the Linear Process in a Hilbert Space under Negative Association

Ko, Mi-Hwa;

Ko, Mi-Hwa;

Abstract

We prove a central limit theorem for the negatively associated random variables in a Hilbert space and extend this result to the linear process generated by negatively associated random variables in a Hilbert space. Our result implies an extension of the central limit theorem for the linear process in a real space under negative association to a simplest case of infinite dimensional Hilbert space.

Keywords

Central limit theorem;negatively associated;linear operator;H-valued random variable;linear process;

Language

English

Cited by

1.

Precise Rates in Complete Moment Convergence for Negatively Associated Sequences,;

References

1.

Araujo, A. and Gine, E. (1980). The Central Limit Theorem for Real and Banach Valued Random Variables, John Wiley & Sons, New York

2.

Billingsley, P. (1968). Convergence of Probability Measures, John Wiley & Sons, New York

3.

Bosq, D. (2000). Linear processes in function spaces, Lectures Notes in Statistics, 149, Springer, Berlin

4.

Brockwell, P. J. and Davis, R. A. (1987). Time Series, Theory and Method, Springer, Berlin

5.

Burton, R. M., Dabrowski, A. R. and Dehling, H. (1986). An invariance principle for weakly associ-ated random vectors, Stochastic Processes and Their Applications, 23, 301-306

6.

Esary, J., Proschan, F. and Walkup, D. (1967). Association of random variables with applications, The Annals of Mathematical Statistics, 38, 1466-1474

7.

Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables, with applications, The Annals of Statistics, 11, 286-295

8.

Kim, T. S. and Ko, M. H. (2008). A central limit theorem for the linear process generated by associated random variables in a Hilbert space, Statistics & Probability Letters, 78, 2102-2109

9.

Kim, T. S., Ko, M. H. and Han, K. H. (2008). On the almost sure convergence for a linear process generated by negatively associated random variables in a Hilbert space, Statistics & Probability Letters, 78, 2110-2115

10.

Ko, M. H. (2006). Functional central limit theorems for multivariate linear processes generated by dependent random vectors, Communications of the Korean Mathematical Society, 21, 779-786

11.

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, Journal of Theoretical Probability, 22, 506-513

12.

Ko, M. H., Ryu, D. H. and Han, K. H. (2006). A central limit theorem for moving average process with negatively associated innovation, International Mathematical Forum, 1, 492-502