We revisit an old problem related to Autoregressive Moving Average (ARMA) models, on quantifying and bounding the approximation error between a true stationary process $X_t$ and an ARMA model $Y_t$. We take the transfer function representation of an ARMA model and show that the associated $L^{\infty}$ norm provides a valid alternate norm that controls the $L^2$ norm and has structural properties comparable to the cepstral norm. We show that a certain subspace of stationary processes, which includes ARMA models, forms a Banach algebra under the $L^{\infty}$ norm that respects the group structure of $H^{\infty}$ transfer functions. 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：本文重新审视了与自回归滑动平均（ARMA）模型相关的一个经典问题：如何量化并界定真实平稳过程$X_t$与ARMA模型$Y_t$之间的逼近误差。我们采用ARMA模型的传递函数表示，证明其对应的$L^{\infty}$范数构成一种有效的替代范数，该范数能够控制$L^2$范数，并具有与倒谱范数相似的结构特性。我们证明，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，且该代数保持$H^{\infty}$传递函数的群结构。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这一简化背景下计算了若干显式逼近界，并对帕德逼近与简约模型中的某些启发式思想进行了评析。