On linear weak predictability with single point spectrum degeneracy

Abstract

The paper studies properties of continuous time processes with spectrum degeneracy at a single point where their Fourier transforms vanish with a certain rate. It appears that these processes are linearly predictable in some weak sense, meaning that convolution integrals over future times can be approximated by causal convolutions over past times. The corresponding predicting kernels are time invariant, and they are presented explicitly in the frequency domain via their transfer functions. These predictors are "universal" meaning that they do not require to know details of the spectrum of the underlying processes; the same predictor can be used for the entire class of processes with a single point spectrum degeneracy. The predictors feature some robustness with respect to noise contamination.