REPRESENTATION OF I(1) AND I(2) AUTOREGRESSIVE HILBERTIAN PROCESSES
将Granger-Johansen表示定理推广到取值于复可分希尔伯特空间的I(1)和I(2)过程，为函数型时间序列的统计应用提供理论基础，并刻画了协整子空间和吸引子子空间。
We develop versions of the Granger–Johansen representation theorems for I(1) and I(2) vector autoregressive processes that apply to processes taking values in an arbitrary complex separable Hilbert space. This more general setting is of central relevance for statistical applications involving functional time series. An I(1) or I(2) solution to an autoregressive law of motion is obtained when the inverse of the autoregressive operator pencil has a pole of first or second order at one. We obtain a range of necessary and sufficient conditions for such a pole to be of first or second order. Cointegrating and attractor subspaces are characterized in terms of the behavior of the autoregressive operator pencil in a neighborhood of one.