ON THE HAJECK-RENYI-TYPE INEQUALITY FOR $\small{\tilde{\rho}}$-MIXING SEQUENCES

• Journal title : Honam Mathematical Journal
• Volume 30, Issue 3,  2008, pp.479-486
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2008.30.3.479
Title & Authors
ON THE HAJECK-RENYI-TYPE INEQUALITY FOR $\small{\tilde{\rho}}$-MIXING SEQUENCES
Choi, Jeong-Yeol; Baek, Jong-Il;

Abstract
Let {$\small{{\Omega}}$, F, P} be a probability space and {$\small{X_n{\mid}n{\geq}1}$} be a sequence of random variables defined on it. We study the Hajeck-Renyi-type inequality for p..mixing random variable sequences and obtain the strong law of large numbers by using this inequality. We also consider the strong law of large numbers for weighted sums of $\small{{\tilde{\rho}}}$-mixing sequences.
Keywords
$\small{{\tilde{\rho}}}$-mixing random variable sequence;Hajeck-Renyi inequality;Strong law of large numbers;
Language
English
Cited by
References
1.
Bradley, R.C., 1990. Equivalent mixing conditions of random fields. Technical Report No. 336. Center for Stochastic Processes, University of North Carolina, Chapel Hill.

2.
Bradley, R.C., 1992. On the spectral density and asymptotic normality of weakly dependent random fields. J. Theoret. Probab. 5, 355-374.

3.
Bryc, W., Smolenski, W., 1993. Moment conditions for almost sure convergence of weakly correlated random variables. Proc. Amer. Math. Soc. 119 (2), 629-635.

4.
Chow, Y.S., 1960. A martingale inequality and the law of large numbers. Proc. Amer. Math. Soc. 11, 107-111.

5.
Christofides, T.C., 2000. Maximal inequalities for demimartingales and a strong law of large number. Statist. & Probab. Lett. 50, 357-363.

6.
Can, Shixin., 1997. The Hajeck-Renyi inequality for Banach space valued mar-tingales and the p smoothness of Banach space, Statist. & Probab. Lett. 32, 245-248.

7.
Gan, Shixin., 2004. Almost sure convergence for $\tilde{\rho}$-mixing random variable sequences. Stat. & Proba. Lett. 67, 289-298.

8.
Hajeck, J. and Renyi, A., 1955. Generalization of an inequality of Kolmogorov, Acta. Math. Acad. Sci. Hungar. 6, 281-283.

9.
Liu, J., Gan, S. and Chen, P., 1999. The Hajeck-Renyi inequality for the NA random variables and its application, Statist. & Probab. Lett. 43, 99-105.

10.
Prakssa Rao, B.L.S., 2002. Hajeck - Renyi - type inequality for associated sequences. Statis. & Proba. Lett. 57, 139-143.

11.
Wu Qunying, 2001. Some convergence properties for $\tilde{\rho}$-mixing sequences. J. Engng. Math. (Chinese) 18 (3), 58-64.

12.
Wu Qunying, 2002. Convergence for weighted sums of $\tilde{\rho}$-mixing random sequences. Math. Appl. (Chinese) 15 (1), 1-4.

13.
Yang Shanchao, 1998. Some moment inequalities for partial sums of random variables and their applications. Chinese Sci. Bull. 43 (17), 1823-1827.