# Recurrent PACF expression on ACF and PACF of lower order

Let $$\rho_{partial}(n) = Cor(Y_t, Y_{t-n}|Y_{t-1}=\mu,\cdots Y_{t-2}=\mu, Y_{t-n+1}=\mu)$$ where $$\mu$$ is the mean of stationary process.

I know that $$\rho_{partial}(1)= \rho(1)$$ and that $$\rho_{partial}(2)=\frac{\rho(2)-\rho^2(1)}{1-\rho^2(1)}$$

I heard that all partial auto-correlations can be recurrently represented with auto-correlations and partial auto-correlations of lower order.

Could you give me a reference and/or explain how partials of higher order can be obtained?

UPD This answer suggests using Durbin-Lewinson rule, that contains a toepliz matrix. Is it true that the matrix entries are somehow special such that no matter how big the matrix is, the answers for $$\rho_{partial(k)}$$ depends only on lower order $$\rho_{partials}$$?

$$\begin{eqnarray} \left(\begin{array}{cccc} \rho(0) & \rho(1) & \cdots & \rho(k-1) \\ \rho(1) & \rho(0) & \cdots & \rho(k-2) \\ \vdots & \vdots & \vdots & \vdots \\ \rho(k-1) & \rho(k-2) & \cdots & \rho(0) \\ \end{array}\right) \left(\begin{array}{c} \phi_{k1} \\ \phi_{k2} \\ \vdots \\ \phi_{kk} \\ \end{array}\right) = \left(\begin{array}{c} \rho(1) \\ \rho(2) \\ \vdots \\ \rho(k) \\ \end{array}\right) \,, \end{eqnarray}$$

UPD2 Some hand-waving We can treat any stationary process as an AR process, and then the partial autocorrelations will be coefficients $$\phi_n$$. When we write $$\rho(k)$$ we get a sum of rhos with coefficients so the matrix arises.