We propose an $L^2$ norm for stationary Autoregressive Moving Average (ARMA) models. We look at ARMA models within the Hilbert space of the past with present of a true purely linearly non-deterministic stationary process $X_t$, and compute the $L^2$ norm based on its Wold decomposition. As an application of this $L^2$ norm, we derive bounds on the mean square prediction error for AR(1) models of MA(1) processes, and verify these bounds empirically for sample data.

翻译：我们针对平稳自回归滑动平均（ARMA）模型提出了一种$L^2$范数。在真实纯线性非确定性平稳过程$X_t$的过去与现在的希尔伯特空间中考察ARMA模型，并基于其Wold分解计算$L^2$范数。作为该$L^2$范数的一个应用，我们推导了MA(1)过程对AR(1)模型的均方预测误差界限，并通过样本数据对这些界限进行了实证验证。