最优因果滤波器有限截段的渐近行为
On the asymptotic behavior of a finite section of the optimal causal filter
AI总结：
本文研究最优因果滤波器有限样本近似的渐近行为，通过精确系数表达式与一致 Baxter 不等式建立短/长记忆二阶平稳时间序列下的 $L_1$ 误差界，用于均方误差预测性能评估。
摘要：
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一个 $L_1$ 界。这里，我们假设数据生成过程是具有短记忆或长记忆自协方差的二阶平稳时间序列。为推导该 $L_1$ 界，我们首先分别依据多步向前无限预测器系数和有限预测器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，我们证明一个所谓的一致 Baxter 不等式，以获得短记忆和长记忆时间序列中无限与有限多步向前预测器系数之差的界。因果滤波器系数的这一 $L_1$ 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。
英文摘要：
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.
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