We look at a problem related to Autoregressive Moving Average (ARMA) models, on quantifying the approximation error between a true stationary process $X_t$ and an ARMA model $Y_t$. We take the transfer function representation $x(L)$ of a stationary process $X_t$ and show that the $L^{\infty}$ norm of $x$ acts as a valid norm on $X_t$ that controls the $\ell^2$ norm of its Wold coefficients. We then show that a certain subspace of stationary processes, which includes ARMA models, forms a Banach algebra under the $L^{\infty}$ norm that respects the multiplicative structure of $H^{\infty}$ transfer functions and thus improves on the structural properties of the cepstral norm for ARMA models. The natural definition of invertibility in this algebra is consistent with the original definition of ARMA invertibility, and generalizes better to non-ARMA processes than Wiener's $\ell^1$ condition. Finally, we calculate some explicit approximation bounds in the simpler context of continuous transfer functions, and critique some heuristic ideas on Pad\'e approximations and parsimonious models.

翻译：本文研究自回归滑动平均（ARMA）模型相关的一个问题：如何量化真实平稳过程$X_t$与ARMA模型$Y_t$之间的逼近误差。我们采用平稳过程$X_t$的传递函数表示$x(L)$，证明$x$的$L^{\infty}$范数可作为$X_t$的有效范数，并能控制其Wold系数的$\ell^2$范数。进一步证明，包含ARMA模型的某类平稳过程子空间在$L^{\infty}$范数下构成Banach代数，该代数保持了$H^{\infty}$传递函数的乘法结构，从而改进了ARMA模型倒谱范数的结构性质。该代数中可逆性的自然定义与ARMA可逆性的原始定义一致，且相较于Wiener的$\ell^1$条件能更好地推广至非ARMA过程。最后，我们在连续传递函数的简化情境下计算了若干显式逼近界，并对帕德逼近与简约模型的相关启发式观点进行了评析。