### Abstract

Let X_{n} = Σ_{i}∞=1 a_{i}ε _{n}-_{i}, where the ε_{i} are i.i.d, with mean 0 and at least finite second moment, and the a_{i} are assumed to satisfy a_{i} = O(_{i}^{-β}) with β > 1/2. When 1/2 < β < 1, X_{n} is usually called a long-range dependent or long-memory process. For a certain class of Borel functions K(x_{1},..., x_{d+1}), d ≥0, from R^{d+1} to R, which includes indicator functions and polynomials, the stationary sequence K (X_{n}, X_{n}+1,..., X_{n+d}) is considered. By developing a finite orthogonal expansion of K (X_{n},..., X_{n+d}), the Berry-Esseen type bounds for the normalized sum Q_{N}/√N, Q_{N} = ε _{n}^{N}=_{1} (K(X_{n},..., X_{n+d}) - EK(X_{n},..., X_{n+d})) are obtained when Q_{N}/√N obeys the central limit theorem with positive limiting variance.

Original language | English |
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Pages (from-to) | 301-321 |

Number of pages | 21 |

Journal | Bernoulli |

Volume | 14 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2008 May 1 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability

### Cite this

*Bernoulli*,

*14*(2), 301-321. https://doi.org/10.3150/07-BEJ112

}

*Bernoulli*, vol. 14, no. 2, pp. 301-321. https://doi.org/10.3150/07-BEJ112

**On Berry-Esseen bounds for non-instantaneous filters of linear processes.** / Cheng, Tsung L.; Ho, Hwai Chung.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On Berry-Esseen bounds for non-instantaneous filters of linear processes

AU - Cheng, Tsung L.

AU - Ho, Hwai Chung

PY - 2008/5/1

Y1 - 2008/5/1

N2 - Let Xn = Σi∞=1 aiε n-i, where the εi are i.i.d, with mean 0 and at least finite second moment, and the ai are assumed to satisfy ai = O(i-β) with β > 1/2. When 1/2 < β < 1, Xn is usually called a long-range dependent or long-memory process. For a certain class of Borel functions K(x1,..., xd+1), d ≥0, from Rd+1 to R, which includes indicator functions and polynomials, the stationary sequence K (Xn, Xn+1,..., Xn+d) is considered. By developing a finite orthogonal expansion of K (Xn,..., Xn+d), the Berry-Esseen type bounds for the normalized sum QN/√N, QN = ε nN=1 (K(Xn,..., Xn+d) - EK(Xn,..., Xn+d)) are obtained when QN/√N obeys the central limit theorem with positive limiting variance.

AB - Let Xn = Σi∞=1 aiε n-i, where the εi are i.i.d, with mean 0 and at least finite second moment, and the ai are assumed to satisfy ai = O(i-β) with β > 1/2. When 1/2 < β < 1, Xn is usually called a long-range dependent or long-memory process. For a certain class of Borel functions K(x1,..., xd+1), d ≥0, from Rd+1 to R, which includes indicator functions and polynomials, the stationary sequence K (Xn, Xn+1,..., Xn+d) is considered. By developing a finite orthogonal expansion of K (Xn,..., Xn+d), the Berry-Esseen type bounds for the normalized sum QN/√N, QN = ε nN=1 (K(Xn,..., Xn+d) - EK(Xn,..., Xn+d)) are obtained when QN/√N obeys the central limit theorem with positive limiting variance.

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UR - http://www.scopus.com/inward/citedby.url?scp=48049121095&partnerID=8YFLogxK

U2 - 10.3150/07-BEJ112

DO - 10.3150/07-BEJ112

M3 - Article

VL - 14

SP - 301

EP - 321

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 2

ER -