COINTEGRATION IN FUNCTIONAL AUTOREGRESSIVE PROCESSES
定义了希尔伯特空间值自回归过程的有限型单位根，推广了Granger-Johansen表示定理到任意整数阶，并刻画了协整空间的结构，对高维时间序列建模有理论价值。
This article defines the class of ${\cal H}$ -valued autoregressive (AR) processes with a unit root of finite type, where ${\cal H}$ is a possibly infinite-dimensional separable Hilbert space, and derives a generalization of the Granger–Johansen Representation Theorem valid for any integration order $d = 1,2, \ldots$ . An existence theorem shows that the solution of an AR process with a unit root of finite type is necessarily integrated of some finite integer order d , displays a common trends representation with a finite number of common stochastic trends, and it possesses an infinite-dimensional cointegrating space when ${\rm{dim}}{\cal H} = \infty$ . A characterization theorem clarifies the connections between the structure of the AR operators and ( i ) the order of integration, ( ii ) the structure of the attractor space and the cointegrating space, ( iii ) the expression of the cointegrating relations, and ( iv ) the triangular representation of the process. Except for the fact that the dimension of the cointegrating space is infinite when ${\rm{dim}}{\cal H} = \infty$ , the representation of AR processes with a unit root of finite type coincides with the one of finite-dimensional VARs, which can be obtained setting ${\cal H} = ^p $ in the present results.