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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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