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

Abstract

Let Xn=i=1aiϵniX_n=\sum_{i=1}^{\infty}a_i\epsilon_{n-i}, where the ϵi\epsilon_i are i.i.d. with mean 0 and at least finite second moment, and the aia_i are assumed to satisfy ai=O(iβ)|a_i|=O(i^{-\beta}) with β>1/2\beta >1/2. When 1/2<β<11/2<\beta<1, XnX_n is usually called a long-range dependent or long-memory process. For a certain class of Borel functions K(x1,...,xd+1)K(x_1,...,x_{d+1}), d0d\ge0, from Rd+1{\mathcal{R}}^{d+1} to R\mathcal{R}, which includes indicator functions and polynomials, the stationary sequence K(Xn,Xn+1,...,Xn+d)K(X_n,X_{n+1},...,X_{n+d}) is considered. By developing a finite orthogonal expansion of K(Xn,...,Xn+d)K(X_n,...,X_{n+d}), the Berry--Esseen type bounds for the normalized sum QN/N,QN=n=1N(K(Xn,...,Xn+d)EK(Xn,...,Xn+d))Q_N/\sqrt{N},Q_N=\sum_{n=1}^N(K(X_ n,...,X_{n+d})-\mathrm{E}K(X_n,...,X_{n+d})) are obtained when QN/NQ_N/\sqrt{N} obeys the central limit theorem with positive limiting variance.

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