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

翻译：我们推导了应用于数据生成过程的最优因果滤波器系数与其基于有限样本观测的逼近之间的$L_1$界。这里，我们假设数据生成过程是二阶平稳的，且具有短记忆或长记忆自协方差。为获得该$L_1$界，我们首先分别给出因果滤波器系数及其逼近的精确表达式，这些表达式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。然后，我们证明一种所谓的均匀型Baxter不等式，以获取两个多步超前预测系数（在短记忆和长记忆时间序列下）之差的界。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预测的质量。