## Bernoulli

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

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

Tsung-Lin Cheng and Hwai-Chung Ho

#### Abstract

Let *X*_{n}=∑^{∞}_{i=1}*a*_{i}*ɛ*_{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 to , 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}_{n=1}(*K*(*X*_{n}, …, *X*_{n+d})−E*K*(*X*_{n}, …, *X*_{n+d})) are obtained when 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.3150/07-BEJ112

**Mathematical Reviews number (MathSciNet)**

MR2544089

**Zentralblatt MATH identifier**

1155.62064

**Keywords**

Berry–Esseen bounds linear processes long memory long-range dependence non-instantaneous filters rate of convergence

#### 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