Limiting behaviour of moving average processes under negative association assumption. (English) Zbl 1199.60074

Chen, P.; Hu, T.-C.; Volodin, A.

Teor. Jmovirn. Mat. Stat. 77, 149-160 (2007) and Theory Probab. Math. Stat. 77, 165-176 (2008).

Let \(\{Y_i,-\infty<i<+\infty\}\) be a doubly infinite sequence of identically distributed negatively associated random variables, and let \(\{a_i,-\infty<i<+\infty\}\) be an absolutely summable sequence of real numbers. Let \(X_n=\sum_{i=-\infty}^{\infty}a_iY_{i+n},n\geq1,\) be the moving average process based on the sequence \(\{Y_i,-\infty<i<+\infty\}\). The authors deal with the convergence and moment convergence of the maximum \(\max_{1\leq k\leq n}| S_k| \) of partial sums \(S_n=\sum_{k=1}^nX_k\) of the moving average processes. Under some moment restrictions they prove that \[ \sum_{n=1}^{\infty}n^{r-2}P\left\{\max_{1\leq k\leq n}| S_k| >\varepsilon n^{1/p}\right\}<\infty \] and \[ \sum_{n=1}^{\infty}n^{r-2-q/p}E\left\{\max_{1\leq k\leq n}| S_k| -\varepsilon n^{1/p}\right\}_+^q<\infty \] for all \(\varepsilon>0\). These results improve the results of J.-I. Baek, T. S. Kim and H. Y. Liang [Aust. N. Z. J. Stat. 45, No. 3, 331–342 (2003; Zbl 1082.60028)] and Y. X. Li and L. X. Zhang [Stat. Probab. Lett. 70, No. 3, 191-197 (2004; Zbl 1056.62100)].

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60F15

Strong limit theorems

Keywords:

moving average process; negatively associated random variables

Citations:

Zbl 1082.60028; Zbl 1056.62100