The Annals of Statistics

On the asymptotic expansion of the empirical process of long-memory moving averages

Hwai-Chung Ho and Tailen Hsing

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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 $Y_{N, r} = \sum_{n=1}^N \sum_{1\leq j_1 < \dots < j_r} \Pi_{s=1}^r a_{j_s}, Y_{N, 0} = N, \sigma_{N, r}^2 = \Var(Y_{N, r})$ 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 $p < (2\beta - 1)^{-1}$, it is shown under mild regularity conditions that, with probability 1, $$\sum_{n=1}^N I(X_n \leq x) = \sum_{r=0}^p (-1)^r F^{(r)} (x) Y_{N,r} + o(N^{-\lambda} \sigma_{N,p})\\ \text{uniformly for all $x \epsilon\Re \forall 0 < \lambda < (\beta - 1/2) \wedge (1/2 - p(\beta - 1/2))$}.$$ This generalizes a host of existing results and provides the vehicle for a number of statistical applications.

Article information

Ann. Statist., Volume 24, Number 3 (1996), 992-1024.

First available in Project Euclid: 20 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G10: Stationary processes
Secondary: 60G30: Continuity and singularity of induced measures 60F17: Functional limit theorems; invariance principles

Asymptotic expansion empirical process fractional ARIMA process long-range dependence noncentral limit theorem


Ho, Hwai-Chung; Hsing, Tailen. On the asymptotic expansion of the empirical process of long-memory moving averages. Ann. Statist. 24 (1996), no. 3, 992--1024. doi:10.1214/aos/1032526953.

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