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

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

Tsung-Lin Cheng and Hwai-Chung Ho

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

Abstract

Let X_n=∑^∞_i=1a_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(x₁, …, 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)−EK(X_n, …, X_n+d)) are obtained when $Q_{N}/\sqrt{N}$ obeys the central limit theorem with positive limiting variance.

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

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.bj/1208872106
Digital Object Identifier: doi:10.3150/07-BEJ112
Mathematical Reviews number (MathSciNet): MR2544089

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