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I am currently reading the Brookwell and Davis Book and cuurently read about the PACF. On page 98 they derive the PACF for the AR(1) model $$ X(t)=0.9X(t-1)+Z(t) $$ and say that the orthogonal projection of $X(1)$ on $\overline{\text{sp}}\{X(2),...,X(k)\}$ is $0.9X(2)$. How does one proof this?

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