Application of the theory of \(\text{KM}_ 2\)O-Langevin equations to the linear prediction problem for the multi-dimensional weakly stationary time series
We recall and rearrange a part of the Wiener-Masani theory in [\textit{N. Wiener} and \textit{P. Masani}, Acta Math. 98, 111-150 (1957; Zbl 0080.130)]. The theory of \(\text{KM}_ 2 \text{O}\)-Langevin equations will be introduced in \S 3 according to [the author, Algebraic analysis, Pap. dedic. to Prof. Mikio Sato on the Occas. of his Sixtieth Birthday, Vol. 2, 601-645 (1989; Zbl 0673.60045)]. In particular, we rearrange the \(\text{KM}_ 2 \text{O}\)-Langevin data associated with the process \(X\) which consists of the triplet of the forward and backward \(\text{KM}_ 2 \text{O}\)-Langevin delay functions, the forward and backward \(\text{KM}_ 2 \text{O}\)-Langevin partial correlation functions, and the forward and backward \(\text{KM}_ 2 \text{O}\)-Langevin fluctuation functions. By taking certain scaling limit of the forward (resp. backward) \(\text{KM}_ 2 \text{O}\)-Langevin equation, we derive in \S 4 a forward (resp. backward) \(\text{AR} (\infty)\)-Langevin equation which governs the time evolution of the time series \(X\). \S 5 treats the global behaviour of the \(\text{KM}_ 2 \text{O}\)-Langevin partial correlation (resp. delay) functions by using the algorithms mentioned above. In \S 6 we obtain two kinds of prediction formulae for the linear predictor in terms of the innovation process \(\varepsilon_ +\) and the process \(X\), respectively. In the final \S 7, we give a concrete representation theorem for the generating function in the Wiener-Masani theory [see \textit{N. Wiener} and \textit{P. Masani}, Acta. Math. 99, 93-137 (1958; Zbl 0080.130) and \textit{P. Masani}, ibid. 104, 141-162 (1960; Zbl 0096.115)] which, according to the nomenclature in the one-dimensional time series, will be called in this paper an outer matrix function of the spectral density matrix function \(\Delta\) of the time series \(X\).
\(\text{KM}_ 2 \text{O}\)-Langevin equations
scientific article; zbMATH DE number 469061