MAXIMAL INEQUALITIES AND AN APPLICATION UNDER A WEAK DEPENDENCE

- Journal title : Journal of the Korean Mathematical Society
- Volume 53, Issue 1, 2016, pp.57-72
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2016.53.1.057

Title & Authors

MAXIMAL INEQUALITIES AND AN APPLICATION UNDER A WEAK DEPENDENCE

HWANG, EUNJU; SHIN, DONG WAN;

HWANG, EUNJU; SHIN, DONG WAN;

Abstract

We establish maximal moment inequalities of partial sums under -weak dependence, which has been proposed by Doukhan and Louhichi [P. Doukhan and S. Louhichi, A new weak dependence condition and application to moment inequality, Stochastic Process. Appl. 84 (1999), 313-342], to unify weak dependence such as mixing, association, Gaussian sequences and Bernoulli shifts. As an application of maximal moment inequalities, a functional central limit theorem is developed for linear processes with -weakly dependent innovations.

Keywords

weak dependence;maximal moment inequality;linear process;functional central limit theorem;

Language

English

References

1.

P. Ango Nze, P. Buhlmann, and P. Doukhan, Weak dependence beyond mixing and asymptotics for nonparametric regression, Ann. Statist. 30 (2002), no. 2, 397-430.

2.

P. J. Bickel and P. Bulmann, A new mixing notion and functional central limit theorems for a sieve bootstrap in time series, Bernoulli 5 (1999), no. 3, 413-446.

3.

J. Dedecker and P. Doukhan, A new covariance inequality and applications, Stochastic Process. Appl. 106 (2003), no. 1, 63-80.

4.

J. Dedecker, P. Doukhan, G. Lang, R. Leon, R. Jose Rafael, S. Louhichi, and C. Prieur, Weak dependence: with examples and applications, Lecture Notes in Statistics, 190, Springer, New York, 2007.

5.

P. Doukhan and S. Louhichi, A new weak dependence condition and applications to moment inequalities, Stochastic Process. Appl. 84 (1999), no. 2, 313-342.

6.

P. Doukhan and M. H. Neumann, Probability and moment inequalities for sums of weakly dependent random variables with applications, Stochastic Process. Appl. 117 (2007), no. 7, 878-903.

7.

E. Hwang and D. W. Shin, A study on moment inequalities under a weak dependence, J. Korean Statist. Soc. 42 (2013), no. 1, 133-141.

8.

S. Lee, Random central limit theorem for the linear process generated by a strong mixing process, Statist. Probab. Lett. 35 (1997), no. 2, 189-196.

9.

B. L. S. Prakasa Rao, Random central limit theorems for martingales, Acta. Math. Acad. Sci. Hungar. 20 (1969), 217-222.

10.

A. Reyni, On the central limit theorem for the sum of a random number of independent random variables, Acta. Math. Acad. Sci. Hungar. 11 (1960), 97-102.

11.

G. G. Roussas and D. A. Ioannides, Moment inequalities for mixing sequences of random variables, Stochastic Anal. Appl. 5 (1987), no. 1, 61-120.

12.

S. Utev and M. Peligrad, Maximal inequalities and an invariance principle for a class of weakly dependent random variables, J. Theoret. Probab. 16 (2003), no. 1, 101-115.

13.

G. Xing, S. Yang, and A. Chen, A maximal moment inequaltiy for $\alpha$ -mixing sequences and its applications, Statist. Probab. Lett. 79 (2009), 1429-1437.

14.

W. Xuejun, H. Shuhe, S. Yan, and Y.Wenzhi, Moment inequality for $\varphi$ -mixing sequences and its applications, J. Inequal. Appl. (2009), Art. ID 379743, 12 pp.