We establish an $L_1$-bound between the coefficients of the optimal causal filter applied to the data-generating process and its finite sample approximation. Here, we assume that the data-generating process is a second-order stationary time series with either short or long memory autocovariances. To derive the $L_1$-bound, we first provide an exact expression for the coefficients of the causal filter and their approximations in terms of the absolute convergent series of the multistep ahead infinite and finite predictor coefficients, respectively. Then, we prove a so-called uniform Baxter's inequality to obtain a bound for the difference between the infinite and finite multistep ahead predictor coefficients in both short and long memory time series. The $L_1$-approximation error bound for the causal filter coefficients can be used to evaluate the performance of the linear predictions of time series through the mean squared error criterion.

翻译：本文建立了数据生成过程所对应的最优因果滤波器系数与其有限样本近似之间的$L_1$范数界。假设数据生成过程为具有短记忆或长记忆自协方差函数的二阶平稳时间序列。为推导该$L_1$范数界，我们首先基于多步超前无限预测器系数与有限预测器系数的绝对收敛级数，分别给出因果滤波器系数及其近似形式的精确表达式。进而证明一种被称为一致Baxter不等式的结果，从而在短记忆和长记忆时间序列中获取无限与有限多步超前预测器系数之差的界。该因果滤波器系数的$L_1$逼近误差界可用于通过均方误差准则评估时间序列线性预测的性能。