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

Let $X_n={\sum }_{i=1}^{\mathrm{\infty }}{a}_i{ϵ}_{n-i}$, 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 $\mathrm{\mid }{a}_i\mathrm{\mid }=O(i^{-\beta })$ with $\beta >1\mathrm{/}2$. When $1\mathrm{/}2<\beta <1$, $X_n$ is usually called a long-range dependent or long-memory process. For a certain class of Borel functions $K(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_{d+1})$, $d\ge 0$, from ${\mathcal{R}}^{d+1}$ to $\mathcal{R}$, which includes indicator functions and polynomials, the stationary sequence $K(X_n,X_{n+1},\mathrm{.}\mathrm{.}\mathrm{.},X_{n+d})$ is considered. By developing a finite orthogonal expansion of $K(X_n,\mathrm{.}\mathrm{.}\mathrm{.},X_{n+d})$, the Berry–Esseen type bounds for the normalized sum $Q_N/\sqrt{N},Q_N=\sum_{n=1}^N(K(X_ n,…,X_{n+d})-$\mathrm{E}$K(X_n,…,X_{n+d}))$areobtainedwhen$Q_N/\sqrt{N}$ obeys the central limit theorem with positive limiting variance.