##
**Abstract**

In this paper, we discuss the moving-average process \( X_{k} = {\sum\nolimits_{i = - \infty }^\infty {\alpha _{{i + k}} \varepsilon _{i} } } \), where {α_{
i
};-∞ < *i* < ∞} is a doubly infinite sequence of identically distributed φ-mixing or negatively associated random variables with mean zeros and finite variances, {α_{
i
};-∞ < *i* < ∞} is an absolutely summable sequence of real numbers. Set \( S_{n} = {\sum\nolimits_{k = 1}^n {X_{k} ,n \geqslant 1} } \). Suppose that \( \sigma ^{2} = E\varepsilon ^{2}_{1} + 2{\sum\nolimits_{k = 2}^\infty {E\varepsilon _{1} \varepsilon _{k} } } > 0 \). We prove that for any \( \delta \geqslant 0,\;{\text{if}}\;E{\left[ {\varepsilon ^{2}_{1} {\left( {\log \;\log {\left| {\varepsilon _{1} } \right|}} \right)}^{{\delta - 1}} } \right]} < \infty \),

, and if \( E{\left[ {\varepsilon ^{2}_{1} {\left( {\log {\left| {\varepsilon _{1} } \right|}} \right)}^{{\delta - 1}} } \right]} < \infty \),

where \( \tau = \sigma \cdot {\sum\nolimits_{i = - \infty }^\infty {\alpha _{i} ,\Gamma {\left( \cdot \right)}} } \) is a Gamma function and μ^{(2δ+2)} stands for the (2δ + 2)-th absolute moment of the standard normal distribution.

### Similar content being viewed by others

## References

Burton, R. M., Dehling, H.: Large deviations for some weakly dependent random process.

*Statist. Probab. Lett.*,**9**, 397–401 (1990)Yang, X. Y.: The law of the iterated logarithm and stochastic index central limit theorem of B-valued stationary linear processes.

*Chin. Ann. of Math.*,**17A**, 703–714 (1996)Li, D. L., Rao, M. B., Wang, X. C.: Complete convergence of moving average processes.

*Statist. Probab. Lett.*,**14**, 111–114 (1992)Zhang, L. X.: Complete convergence of moving average processes under dependence assumptions.

*Statist. Probab. Lett.*,**30**, 165–170 (1996)Gut, A., Spătaru, A.: Precise asymptotics in the law of the iterated logarithm.

*Ann. Probab.*,**28**, 1870–1883 (2000b)Gut, A., Spătaru, A.: Precise asymptotics in the Baum–Katz and Davis laws of large numbers.

*Jour. Math. Anal. Appl.*,**248**, 233–246 (2000a)Shao, Q. M.: A moment inequality and its application.

*Acta Math. Sinica, Chinese Series*,**31**, 736–747 (1988)Shao, Q. M.: Almost sure invariance principles for mixing sequences of random variables.

*Stochastic Processes Appl.*,**48**, 319–334 (1993)Kim, T. S., Baek, J. I.: A central limit theorem for stationary linear processes generated by linearly positively quadrant-dependent process.

*Statist. Probab. Lett.*,**51**, 299–305 (2001)Zhang, L. X., Precise rates in the law of the iterated logarithm, Manuscript, 2001

Zhang, L. X.: Some limit theorems on the law of the iterated logarithm of NA sequences.

*Acta Math. Sinica, Chinese Series*,**47**(3), 541–552 (2004)Su, C., Zhao, L. C., Wang, Y. B.: The moment inequalities and weak convergence for negatively associated sequences. Science in China, 40A, 172–182 (1997)

Billingsley, P.: Convergence of Probability Measures, Wiley, New York, 1968

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

Research supported by National Natural Science Foundation of China

## Rights and permissions

## About this article

### Cite this article

Li, Y.X., Zhang, L.X. Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes.
*Acta Math Sinica* **22**, 143–156 (2006). https://doi.org/10.1007/s10114-005-0542-4

Received:

Revised:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10114-005-0542-4