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- W2019433305 abstract "Let $X_n = Sigma_{i=1}^{infty} a_i varepsilon_{n-i}$, where the $varepsilon_i$ are iid with mean 0 finite fourth moment and the $a_i$ are regularly varying with index $-beta$ where $beta epsilon (1/2, 1)$ so that ${X_n}$ has long-range dependence. This covers an important class of the fractional ARIMA process. For $r geq 0$, let $Y_{N, r} = sum_{n=1}^N sum_{1leq j_1 < dots < j_r} Pi_{s=1}^r a_{j_s}, Y_{N, 0} = N, sigma_{N, r}^2 = Var(Y_{N, r})$ and $F^{(r)} =$ the rth derivative of the distribution function of $X_n$. The $Y_{N, r}$ are uncorrelated and are stochastically decreasing in r. For any positive integer $p < (2beta - 1)^{-1}$, it is shown under mild regularity conditions that, with probability 1, $$sum_{n=1}^N I(X_n leq x) = sum_{r=0}^p (-1)^r F^{(r)} (x) Y_{N,r} + o(N^{-lambda} sigma_{N,p}) text{uniformly for all $x epsilonRe forall 0 < lambda < (beta - 1/2) wedge (1/2 - p(beta - 1/2))$}.$$ This generalizes a host of existing results and provides the vehicle for a number of statistical applications." @default.
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- W2019433305 title "On the asymptotic expansion of the empirical process of long-memory moving averages" @default.
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- W2019433305 doi "https://doi.org/10.1214/aos/1032526953" @default.
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