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.

中文翻译：

---

I(1) 和 I(2) 自回归希尔伯过程的表示

我们为 I(1) 和 I(2) 向量自回归过程开发了 Granger-Johansen 表示定理的版本，适用于在任意复杂的可分离希尔伯特空间中取值的过程。这种更一般的设置与涉及功能时间序列的统计应用程序具有中心相关性。当自回归算子铅笔的倒数具有一阶或二阶极点时，可获得自回归运动定律的 I(1) 或 I(2) 解。我们得到了这样一个极点是一阶或二阶的一系列必要和充分条件。协整和吸引子空间的特征在于自回归算子铅笔在 1 的邻域中的行为。