On the Approximation of Stationary Processes using the ARMA Model

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.