## Abstract

We present some exponential inequalities for positively associated unbounded random variables. By these inequalities, we obtain the rate of convergence *n*
^{−1/2}
*β*
_{
n
}log ^{3/2}
*n* in which *β*
_{
n
} can be particularly taken as (log log *n*)^{1/σ} with any *σ*>2 for the case of geometrically decreasing covariances, which is faster than the corresponding one *n*
^{−1/2}(log log *n*)^{1/2}log ^{2}
*n* obtained by Xing, Yang, and Liu in J. Inequal. Appl., doi:10.1155/2008/385362 (2008) for the case mentioned above, and derive the convergence rate *n*
^{−1/2}
*β*
_{
n
}log ^{1/2}
*n* for the above *β*
_{
n
} under the given covariance function, which improves the relevant one *n*
^{−1/2}(log log *n*)^{1/2}log *n* obtained by Yang and Chen in Sci. China, Ser. A 49(1), 78–85 (2006) for associated uniformly bounded random variables. In addition, some moment inequalities are given to prove the main results, which extend and improve some known results.

### Similar content being viewed by others

## References

Barlow, R.E., Proschan, F.: Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York (1975)

Birkel, T.: Moment bounds for associated sequences. Ann. Probab.

**16**, 1184–1193 (1988)Birkel, T.: On the convergence rate in the central limit theorem for associated processes. Ann. Probab.

**16**, 1685–1698 (1988)Birkel, T.: A note on the strong law of large numbers for positively dependent random variables. Stat. Probab. Lett.

**7**, 17–20 (1989)Cox, J.T., Grimmett, G.: Central limit theorem for associated random variables and the percolation model. Ann. Probab.

**12**, 514–528 (1984)Dewan, I., Prakasa, R.: A general method of density estimation for associated random variables. J. Nonparametr. Stat.

**10**, 405–420 (1999)Esary, J.D., Proschan, F., Walkup, D.W.: Association of random variables, with applications. Ann. Math. Stat.

**38**, 1466–1474 (1967)Ioannides, D.A., Roussas, G.G.: Exponential inequality for associated random variables. Stat. Probab. Lett.

**42**, 423–431 (1999)Joag-Dev, K., Proschan, F.: Negative association of random variables with applications. Ann. Stat. (1983)

Newman, C.M.: Normal fluctuations and the FKG inequalities. Commun. Math. Phys.

**74**, 119–128 (1980)Newman, C.M.: Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Tong, Y.L. (ed.) Inequalities in Statistics and Probability. IMS Lecture Notes—Monograph Series, vol. 5, pp. 127–140. Hayward (1984)

Newman, C.M.: Ising models and dependent percolation. In: Block, H.W., Sampson, A.R., Savits, T.H. (eds.) Topics in Statistical Dependence. IMS Lecture Notes—Monograph Series, vol. 16, pp. 395–401 (1990)

Newman, C.M., Wright, A.L.: Associated random variables and martingale inequalities. Z. Wahrsch. Verw. Geb.

**56**, 361–371 (1982)Oliveira, P.D.: An exponential inequality for associated variables. Stat. Probab. Lett.

**73**, 189–197 (2005)Roussas, G.G.: Kernel estimates under association: Strong uniform consistency. Stat. Probab. Lett.

**12**, 393–403 (1991)Roussas, G.G.: Curve estimation in random fields of associated processes. J. Nonparametr. Stat.

**2**, 215–224 (1993)Roussas, G.G.: Asymptotic normality of random fields of positively or negatively associated processes. J. Multivar. Anal.

**50**, 152–173 (1994)Roussas, G.G.: Asymptotic normality of a smooth estimate of a random field distribution function under association. Stat. Probab. Lett.

**24**, 77–90 (1995)Roussas, G.G.: Exponential probability inequalities with some applications. In: Ferguson, T.S., Shapely, L.S., MacQueen, J.B. (eds.) Statistics, Probability and Game Theory. IMS Lecture Notes—Monograph Series, vol. 30, pp. 303–319. Hayward (1996)

Shao, Q., Yu, H.: Weak convergence for weighted empirical process of dependent sequences. Ann. Probab.

**24**, 2098–2127 (1996)Xing, G., Yang, S., Liu, A.: Exponential inequalities of positively associated random variables and applications. J. Inequal. Appl. (2008). doi:10.1155/2008/385362

Yang, S.: Complete convergence for sums of positively associated sequences. Chin. J. Appl. Probab. Stat.

**17**(2), 197–202 (2001) (in Chinese)Yang, S., Chen, M.: Exponential inequalities for associated random variables and strong law of large numbers. Sci. China Ser. A

**49**(1), 75–85 (2006)

## Author information

### Authors and Affiliations

### Corresponding author

## Rights and permissions

## About this article

### Cite this article

Xing, G., Yang, S. Some Exponential Inequalities for Positively Associated Random Variables and Rates of Convergence of the Strong Law of Large Numbers.
*J Theor Probab* **23**, 169–192 (2010). https://doi.org/10.1007/s10959-008-0205-3

Received:

Revised:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10959-008-0205-3