Open Access
October 1997 Limit theorems for functionals of moving averages
Hwai-Chung Ho, Tailen Hsing
Ann. Probab. 25(4): 1636-1669 (October 1997). DOI: 10.1214/aop/1023481106


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.


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Hwai-Chung Ho. Tailen Hsing. "Limit theorems for functionals of moving averages." Ann. Probab. 25 (4) 1636 - 1669, October 1997.


Published: October 1997
First available in Project Euclid: 7 June 2002

zbMATH: 0903.60018
MathSciNet: MR1487431
Digital Object Identifier: 10.1214/aop/1023481106

Primary: 60F05
Secondary: 60G10

Keywords: asymptotic expansion , central limit theorem , fractional ARIMA process , long memory , long-range dependence , noncentral limit theorem , nonlinear function

Rights: Copyright © 1997 Institute of Mathematical Statistics

Vol.25 • No. 4 • October 1997
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