Cointegrated Linear Processes in Hilbert Space
将多元时间序列的协整概念推广到取值于希尔伯特空间的无限维情形，定义了协整空间并证明了Granger-Johansen表示定理的版本。
We extend the notion of cointegration for multivariate time series to a potentially infinite‐dimensional setting in which our time series takes values in a complex separable Hilbert space. In this setting, standard linear processes with nonzero long‐run covariance operator play the role of processes. We show that the cointegrating space for an process may be sensibly defined as the kernel of the long‐run covariance operator of its difference. The inner product of an process with an element of its cointegrating space is a stationary complex‐valued process. Our main result is a version of the Granger–Johansen representation theorem: we obtain a geometric reformulation of the Johansen I(1) condition that extends naturally to a Hilbert space setting, and show that an autoregressive Hilbertian process satisfying this condition, and possibly also a compactness condition, admits an representation.