Autocorrelation Matrix is Positive Definite

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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 $\sequence {s_n}$.


Then $\mathbf P_n$ is a positive definite matrix.


Proof


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