Determinant of Autocorrelation Matrix is Strictly Positive

Theorem

Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$.

Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$:

$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$

Let $\mathbf P_n$ denote the autocorrelation matrix associated with $S$ for $\sequence {s_n}$.

The determinant of $\mathbf P_n$ is strictly positive.

Proof

We have that the Autocorrelation Matrix is Positive Definite.

The result follows from Determinant of Positive Definite Matrix is Strictly Positive.

$\blacksquare$

Examples

Order $2$

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

Then:

$-1 < \rho_1 < 1$

Order $3$

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

Then:

$-1 < \rho_1 < 1$

$-1 < \rho_2 < 1$

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

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