Second Order Weakly Stationary Gaussian Stochastic Process is Strictly Stationary
Jump to navigation
Jump to search
Theorem
Let $S$ be a Gaussian stochastic process giving rise to a time series $T$.
Let $S$ be weakly stationary of order $2$.
Then $S$ is strictly stationary.
Proof
By definition of a Gaussian process, the probability distribution of $T$ be a multivariate Gaussian distribution.
By definition, a Gaussian distribution is characterized completely by its expectation and its variance.
That is, its $1$st and $2$nd moments.
The result follows.
$\blacksquare$
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: Weak stationarity
- $2.1$ Autocorrelation Properties of Stationary Models:
- $2$: Autocorrelation Function and Spectrum of Stationary Processes:
- Part $\text {I}$: Stochastic Models and their Forecasting: