On the Approximability of Stationary Processes using the ARMA Model

We view the ARMA (Autoregressive Moving Average) model of a stationary process as a random variable approximation. By mapping stationary random variables to Hardy space functions on the unit disk, we can turn a problem of random variable approximation to a newly formulated problem of function approximation. When the functions are continuous, the spectral theorem provides a link between these two points of view, allowing us to provide approximation guarantees for a certain class of stationary processes, and also to identify certain other stationary processes that seem difficult to approximate. We were unable to find similar approximation or approximability guarantees in our review of time series and distributed lags literature. For instance, as detailed in the next section, Box and Jenkins ([BJ76]) assume that a long moving average representation obtained from Wold's decomposition can be mapped to a short autoregressive representation based on some examples and analogies, and provide no specific guarantees for general stationary processes. They tackle the existence of a stable ARMA model using the notion of unit roots.