On the Approximability of Stationary Processes using the ARMA Model

We identify certain gaps in the literature on the approximability of stationary random variables using the Autoregressive Moving Average (ARMA) model. To quantify approximability, we propose that an ARMA model be viewed as an approximation of a stationary random variable. We map these stationary random variables to Hardy space functions, and formulate a new function approximation problem that corresponds to random variable approximation, and thus to ARMA. Based on this Hardy space formulation we identify a class of stationary processes where approximation guarantees are feasible. We also identify an idealized stationary random process for which we conjecture that a good ARMA approximation is not possible. Next, we provide a constructive proof that Pad\'e approximations do not always correspond to the best ARMA approximation. Finally, we note that the spectral methods adopted in this paper can be seen as a generalization of unit root methods for stationary processes even when an ARMA model is not defined.