The Annals of Probability

Limit theorems for functionals of moving averages

Hwai-Chung Ho and Tailen Hsing

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Let $X_n =\sum_{i=1}^\infty a_i \varepsilon_{n-i}$, where the $\varepsilon_i$ are i.i.d. with mean 0 and finite second moment and the $a_i$ are either summable or regularly varying with index $\in (-1,-1/2)$ . The sequence ${X_n}$ has short memory in the former case and long memory in the latter. For a large class of functions $K$, a new approach is proposed to develop both central ($\sqrt{N}$ rate) and noncentral (non-$\sqrt{N}$ rate) limit theorems for $S_N \equiv \sum_{n=1}^N [K(X_n) - EK (X_n)]$. Specifically, we show that in the short-memory case the central limit theorem holds for $S_N$ and in the long-memory case, $S_N$ can be decomposed into two asymptotically uncorrelated parts that follow a central limit and a non-central limit theorem, respectively. Further we write the noncentral part as an expansion of uncorrelated components that follow noncentral limit theorems. Connections with the usual Hermite expansion in the Gaussian setting are also explored.

Article information

Ann. Probab., Volume 25, Number 4 (1997), 1636-1669.

First available in Project Euclid: 7 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G10: Stationary processes

Asymptotic expansion central limit theorem fractional ARIMA process long memory long-range dependence noncentral limit theorem nonlinear function


Ho, Hwai-Chung; Hsing, Tailen. Limit theorems for functionals of moving averages. Ann. Probab. 25 (1997), no. 4, 1636--1669. doi:10.1214/aop/1023481106.

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