Autocovariance Matrix is Positive Definite

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 $\boldsymbol \Gamma_n$ denote the autocovariance matrix associated with $S$ for $\sequence {s_n}$.

Then $\boldsymbol \Gamma_n$ is a positive definite matrix.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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