On a Stochastic Difference Equation for the Multi-Dimensional Weakly Stationary Process with Discrete Time

Abstract

Research in the problem of linear prediction of a one-dimensional weakly stationary process X is classified according to whether the length of the prediction interval is infinitely long. Besides a traditional spectral theory, Krein has developed the Krein's theory for the inverse spectral problem. This chapter explains the problem of linear prediction for X from the viewpoint of the theory of stochastic differential equations. Hyperfunctions play an important role in the study of the infinitely multiple Markovian property. It is important not only from the point of view of statistical physics but also from a probabilistic point of view to derive a stochastic equation of motion describing the time evolution of X. The process X with reflection positivity has been investigated in detail with the object of clarifying a mathematical structure of the fluctuation–dissipation theorem in statistical physics. In the course of these investigations, it has been found that the time evolution of X is described by two kinds of Langevin equations with a notable difference in character of random forces: one is the first KMO-Langevin equation having a white noise as a random force and the other is the second KMO-Langevin equation where a colored noise, called the Kubo noise, is taken to be a random force. It is a key to obtain the structure theorem of the outer function of X, because the Fourier transform of the outer function gives the canonical representation kernel for X.