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 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：我们建立了应用于数据生成过程的最优因果滤波器系数与其有限样本近似之间的$L_1$界。此处假设数据生成过程为具有短记忆或长记忆自协方差的二阶平稳时间序列。为推导$L_1$界，我们首先分别基于多步超前无限预测系数和有限预测系数的绝对收敛级数，给出因果滤波器系数及其近似的精确表达式。随后证明一类所谓的统一巴克斯特不等式，以获取短记忆与长记忆时间序列中无限与有限多步超前预测系数之差的界。该因果滤波器系数的$L_1$近似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。