## On the asymptotic expansion of the empirical process of long-memory moving averages.(English)Zbl 0862.60026

Summary: Let $$X_n=\sum^\infty_{i=1}a_i\varepsilon_{n-i}$$, where the $$\varepsilon_i$$ are i.i.d. with mean 0 and finite fourth moment and the $$a_i$$ are regularly varying with index $$-\beta$$ where $$\beta\in(1/2,1)$$ so that $$\{X_n\}$$ has long-range dependence. This covers an important class of the fractional ARIMA process. For $$r\geq0$$, let $Y_{N,r}= \sum^N_{n=1} \sum_{1\leq j_1<\cdots<j_r}\prod^r_{s=1} a_{j_s}\varepsilon_{n-j_s},\quad Y_{N,0}=N,\quad \sigma^2_{N,r}=\text{Var}(Y_{N,r})$ and $$F^{(r)}=$$ the $$r$$th derivative of the distribution function of $$X_n$$. The $$Y_{N,r}$$ are uncorrelated and are stochastically decreasing in $$r$$. For any positive integer $$p<(2\beta-1)^{-1}$$, it is shown under mild regularity conditions that, with probability 1, $\sum^N_{n=1} I(X_n\leq x)= \sum^p_{r=0} (-1)^rF^{(r)}(x)Y_{N,r}+o(N^{-\lambda}\sigma_{N,p})$ uniformly for all $$x\in{\mathfrak R}$$, $$\forall 0<\lambda<(\beta-1/2)\wedge(1/2-p(\beta-1/2))$$. This generalizes a host of existing results and provides the vehicle for a number of statistical applications.

### MSC:

 60G10 Stationary stochastic processes 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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### References:

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