Determinant of Autocorrelation Matrix is Strictly Positive/Examples/Order 3

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Example of Use of Determinant of Autocorrelation Matrix is Strictly Positive

Let $\rho_k$ denote the autocorrelation of a strictly stationary stochastic process $S$ at lag $1k$.

Then:

$-1 < \rho_1 < 1$
$-1 < \rho_2 < 1$
$-1 < \dfrac {\rho_2 - \rho_1^2} {1 - \rho_1^2} < 1$


Proof

Consider the autocorrelation matrix of order $3$:

\(\ds \map \det {\mathbf P_3}\) \(>\) \(\ds 0\) Determinant of Autocorrelation Matrix is Strictly Positive
\(\ds \begin {vmatrix} 1 & \rho_1 & \rho_2 \\ \rho_1 & 1 & \rho_1 \\ \rho_2 & \rho_1 & 1 \end {vmatrix}\) \(>\) \(\ds 0\) Definition of Autocorrelation Matrix
\(\ds \paren {1 - \rho_1^2} - \rho_1^2 \paren {1 - \rho_2} + \rho_2 \paren {\rho_1^2 - \rho_2}\) \(>\) \(\ds 0\) Definition of Determinant
\(\ds \leadsto \ \ \) \(\ds \) \(<\) \(\ds 1\) irritating algebra that I can't resolve
\(\ds \leadsto \ \ \) \(\ds \size {\dfrac {\rho_2 - \rho_1^2} {1 - \rho_1^2} }\) \(<\) \(\ds 1\)


Also from:

\(\ds \begin {vmatrix} 1 & \rho_1 \\ \rho_1 & 1 \end {vmatrix}\) \(>\) \(\ds 0\)
\(\ds \begin {vmatrix} 1 & \rho_2 \\ \rho_2 & 1 \end {vmatrix}\) \(>\) \(\ds 0\)

the other conditions follow from the order $2$ case.

$\blacksquare$


Sources

Part $\text {I}$: Stochastic Models and their Forecasting:
$2$: Autocorrelation Function and Spectrum of Stationary Processes:
$2.1$ Autocorrelation Properties of Stationary Models:
$2.1.3$ Positive Definiteness and the Autocovariance Matrix: Conditions satisfied by the autocorrelations of a stationary process