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
- 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next):
- 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
- $2.1$ Autocorrelation Properties of Stationary Models:
- $2$: Autocorrelation Function and Spectrum of Stationary Processes:
- Part $\text {I}$: Stochastic Models and their Forecasting: