## Complete convergence of moving average processes under dependence assumptions.(English)Zbl 0873.60019

The author considers a moving-average process $$X_k=\sum_{i=-\infty}^\infty a_{i+k}Y_i$$, $$k\geq 1$$, for an absolutely summable sequence $$\{a_i\}$$ of real numbers and a doubly infinite sequence $$\{Y_i\}$$ of identically distributed and $$\varphi$$-mixing random variables. Under a slight condition on the $$\varphi$$-mixing coefficient and $$EY_1=0$$, $$E|Y_1|^{rt}<\infty$$ for $$1\leq t<2$$, $$r\geq 1$$, the convergence of the series $$\sum_{n=1}^\infty n^{r-2}P(|\sum_{k=1}^n X_k|\geq n^{1/t}\varepsilon)$$ for all $$\varepsilon>0$$ could be proved.

### MSC:

 60F15 Strong limit theorems 60G50 Sums of independent random variables; random walks 60G05 Foundations of stochastic processes

### Keywords:

complete convergence; phi-mixing; moving-average process
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### References:

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