标题： 关于线性过程的非瞬时滤波器的Berry--Esseen界 显示英文标题

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

摘要： 设 $X_n=\sum_{i=1}^{\infty}a_i\epsilon_{n-i}$，其中 $\epsilon_i$是独立同分布的，均值为 0 且至少具有有限的二阶矩，而 $a_i$被假定满足 $|a_i|=O(i^{-\beta})$，其中 $\beta >1/2$。当 $1/2<\beta<1$时， $X_n$通常被称为长程依赖或长记忆过程。 对于一类Borel函数$K(x_1,...,x_{d+1})$,$d\ge0$，从${\mathcal{R}}^{d+1}$到$\mathcal{R}$，其中包括指示函数和多项式，考虑平稳序列$K(X_n,X_{n+1},...,X_{n+d})$。 通过发展$K(X_n,...,X_{n+d})$的有限正交展开，当$Q_N/\sqrt{N}$满足具有正极限方差的中心极限定理时，得到了归一化和$Q_N/\sqrt{N},Q_N=\sum_{n=1}^N(K(X_ n,...,X_{n+d})-\mathrm{E}K(X_n,...,X_{n+d}))$的 Berry--Esseen 型界。 显示英文摘要

摘要： Let $X_n=\sum_{i=1}^{\infty}a_i\epsilon_{n-i}$, where the $\epsilon_i$ are i.i.d. with mean 0 and at least finite second moment, and the $a_i$ are assumed to satisfy $|a_i|=O(i^{-\beta})$ with $\beta >1/2$. When $1/2<\beta<1$, $X_n$ is usually called a long-range dependent or long-memory process. For a certain class of Borel functions $K(x_1,...,x_{d+1})$, $d\ge0$, 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 $Q_N/\sqrt{N},Q_N=\sum_{n=1}^N(K(X_ n,...,X_{n+d})-\mathrm{E}K(X_n,...,X_{n+d}))$ are obtained when $Q_N/\sqrt{N}$ obeys the central limit theorem with positive limiting variance.