# 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

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