Moving-average representation of autoregressive approximations

We study the properties of an MA(∞)-representation of an autoregressive approximation for a stationary, real-valued process. In doing so we give an extension of Wiener's theorem in the deterministic approximation setup. When dealing with data, we can use this new key result to obtain insight into the structure of MA(∞)-representations of fitted autoregressive models where the order increases with the sample size. In particular, we give a uniform bound for estimating the moving-average coefficients via autoregressive approximation being uniform over all integers.