## Bernoulli

• Bernoulli
• Volume 14, Number 2 (2008), 301-321.

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

#### Abstract

Let $X_n=∑^∞_{i=1}a_iɛ_{n−i}$, where the $ɛ_i$ 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 ${\mathcal{R}}^{d+1}$ to $\mathcal{R}$, 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 $Q_{N}/\sqrt{N}, Q_N=∑^N_{n=1}(K(X_n, …, X_{n+d})−\mathrm{E}K(X_n, …, X_{n+d}))$ are obtained when $Q_{N}/\sqrt{N}$ obeys the central limit theorem with positive limiting variance.

#### Article information

Source
Bernoulli, Volume 14, Number 2 (2008), 301-321.

Dates
First available in Project Euclid: 22 April 2008

https://projecteuclid.org/euclid.bj/1208872106

Digital Object Identifier
doi:10.3150/07-BEJ112

Mathematical Reviews number (MathSciNet)
MR2544089

Zentralblatt MATH identifier
1155.62064

#### Citation

Cheng, Tsung-Lin; Ho, Hwai-Chung. On Berry–Esseen bounds for non-instantaneous filters of linear processes. Bernoulli 14 (2008), no. 2, 301--321. doi:10.3150/07-BEJ112. https://projecteuclid.org/euclid.bj/1208872106

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