For AR(P) process, I once read the following property:
P corresponds to the projection on the space $\bar{sp}\{x_2,...x_k\}$, which is defined as follows
I do not know how does equation (3.4.1) was derived
2026-03-29 08:44:48.1774773888
a representation question on AR(P) process
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The RHS of (3.4.1) belongs to the span, $X_t$ equals the RHS of (3.4.1) plus $Z_t$, and the projection of $Z_t$ on the span is $0$. QED.