On the Convergence of Finite Order Approximations of Stationary Time Series

TL;DR

This paper investigates the asymptotic behavior of spectral densities in finite order AR and MA approximations of stationary time series, showing convergence properties under certain conditions.

Contribution

It provides new theoretical results on the convergence of spectral densities of finite order AR and MA approximations for stationary time series.

Findings

Spectral density of AR approximations converges at the origin.

Spectral densities of AR and MA approximations converge in L2 as order increases.

Convergence holds under conditions of non-vanishing spectral density and summable covariance.

Abstract

The approximation of a stationary time-series by finite order autoregressive (AR) and moving averages (MA) is a problem that occurs in many applications. In this paper we study asymptotic behavior of the spectral density of finite order approximations of wide sense stationary time series. It is shown that when the on the spectral density is non-vanishing in $[-\pi,\pi]$ and the covariance is summable, the spectral density of the approximating autoregressive sequence converges at the origin. Under additional mild conditions on the coefficients of the Wold decomposition it is also shown that the spectral densities of both moving average and autoregressive approximations converge in $L_2$ as the order of approximation increases.