On the Convergence of Finite Linear Predictors of Long-memory Stationary Processes

• Activity Number: 179
• Type: Contributed
• Date/Time: Tuesday, August 13, 2002 : 8:30 AM to 10:20 AM
• Sponsor: General Methodology
• Author(s): Pascal Bondon*+ and Wilfredo Palma
• Affiliation(s): CNRS and Pontificia Universidad Catolica de Chile
• Keywords: stationary process ; linear predictor ; long-memory ; ARFIMA model ; rate of convergence

Abstract

We consider the rate with which the best linear predictor based on a finite number of past observations for a univariate stationary process converges to the best linear predictor given the entire infinite past, as the sample size goes to infinity. This problem has been considered by U. Grenander and M. Rosenblatt [Trans. Amer. Math. Soc. 76 (1954)] for a process whose spectral density coincides in $[-\pi,\pi]$ almost everywhere with a strictly positive analytic function, and in that case the difference between the two predictors converges to zero at an exponential rate. The spectral densities of long-range-dependent processes have a pole at zero frequency and therefore do not satisfy the above condition. For these processes, we will establish that the difference between the two predictors converges to zero at a hyperbolic rate.