Predictability of band-limited, high-frequency, and mixed processes in the presence of ideal low-pass filters

Pathwise predictability of continuous time processes is studied in deterministic setting. We discuss uniform prediction in some weak sense with respect to certain classes of inputs. More precisely, we study possibility of approximation of convolution integrals over future time by integrals over past time. We found that all band-limited processes are predictable in this sense, as well as high-frequency processes with zero energy at low frequencies. It follows that a process of mixed type still can be predicted if an ideal low-pass filter exists for this process.Comment: 10 page

On predictors for band-limited and high-frequency time series

Pathwise predictability and predictors for discrete time processes are studied in deterministic setting. It is suggested to approximate convolution sums over future times by convolution sums over past time. It is shown that all band-limited processes are predictable in this sense, as well as high-frequency processes with zero energy at low frequencies. In addition, a process of mixed type still can be predicted if an ideal low-pass filter exists for this process.Comment: 10 pages. arXiv admin note: text overlap with arXiv:0708.034

Predictability on finite horizon for processes with exponential decrease of energy on higher frequencies

The paper presents sufficient conditions of predictability for continuous time processes in deterministic setting. We found that processes with exponential decay on energy for higher frequencies are predictable in some weak sense on some finite time horizon defined by the rate of decay. Moreover, this predictability can be achieved uniformly over classes of processes. Some explicit formulas for predictors are suggested.Comment: 11 page

Predictors for discrete time processes with energy decay on higher frequencies

The predictability of discrete-time processes is studied in a deterministic setting. A family of one-step-ahead predictors is suggested for processes of which the energy decays at higher frequencies. For such processes, the prediction error can be made arbitrarily small. The predictions can be robust with respect to the noise contamination at higher frequencies.Comment: 5 page