Determinant of Autocorrelation Matrix is Strictly Positive/Examples/Order 2

Example of Use of Determinant of Autocorrelation Matrix is Strictly Positive

Let $\rho_1$ be the autocorrelation of a strictly stationary stochastic process $S$ at lag $1$.

Then:

$-1 < \rho_1 < 1$

Proof

Consider the autocorrelation matrix of order $2$:

\(\ds \map \det {\mathbf P_2}\)

\(>\)

\(\ds 0\)

Determinant of Autocorrelation Matrix is Strictly Positive

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

\(>\)

\(\ds 0\)

Definition of Autocorrelation Matrix

\(\ds 1 - \rho_1^2\)

\(>\)

\(\ds 0\)

Definition of Determinant

\(\ds \leadsto \ \ \)

\(\ds \rho_1^2\)

\(<\)

\(\ds 1\)

\(\ds \leadsto \ \ \)

\(\ds \size {\rho_1}\)

\(<\)

\(\ds 1\)

$\blacksquare$

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