## Complete moment convergence of moving average processes with dependent innovations.(English)Zbl 1152.60029

The result of Li and Zhang is extended to the sequence $$\{Y_i;\;-\infty<i<\infty\}$$ of identically distributed and $$\phi-$$mixing r.v. with $$EY_1=0,$$ $$EY_1^2<\infty,$$ satisfying the condition $$\sum_{m=1}^{\infty}\phi^{\frac 12}(m)<\infty,$$ where $$\phi(m)=\sup \phi(\mathcal{F}^k_{\infty},\mathcal{F}_{k+m}^{\infty})\to 0$$ as $$m\to \infty,$$ $$\mathcal{F}_n^m=\sigma(Y_k;\;n\leq k\leq m),$$ $$\phi(A,B)=\sup(P(B|\,A)-P(B)),$$ $$A\in \mathcal{A},B\in \mathcal{B},\;P(A)>0.$$
The main result (theorem 2.3) is formulated as follows. Set $$S_n=\sum_{k=1}^nX_k,$$ where $$X_k=\sum_{i=-\infty}^{\infty}a_{i+k}Y_i,\;k\geq 1,$$ and $$\{a_i;\;-\infty<i<\infty\}$$ is a sequence of real numbers with $$\sum_{i=-\infty}^{\infty}|a_i|<\infty.$$ Let $$h(x)>0 (x>0)$$ be a slowly varying function and $$1\leq p<2,\;r>p.$$ Then $$E|Y_1|^rh(|Y_1|^p)<\infty$$ implies $\sum_{n=1}^{\infty} n^{r/p-2-1/p}h(n)E\{|S_n|-\epsilon n^{1/p}\}^+<\infty \text{\;for\;all}\;\epsilon>0.$

### MSC:

 60F15 Strong limit theorems 60G50 Sums of independent random variables; random walks 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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