On the stationarity of Gaussian processes

Question

I am trying to understand and prove the statement:

The normal (or Gaussian) process is stationary in the wide sense if and only if it is strictly stationary.

I know the following:

1. A strictly stationary process is also stationary in the wide sense if ${E}[x_t^2] < \infty$
2. For a Gaussian process, the entire distribution is determined by the mean and covariance function, making it unique in terms of properties of stationarity.

However, I am struggling with how to formally show this equivalence for Gaussian processes.

Could someone guide me through the proof or provide insights into why this equivalence holds in the Gaussian case?

Thank you!

Answer 1

Suppose that $(X_t)_{t\in\Bbb R}$ is a Gaussian process stationary in the wide sense, so that $m(t):=EX_t=m$ and $Cov\,(X_s,X_t)=g(s-t)$ for some real $m$, some real-valued function $g$, and all real $s$ and $t$.

Then for all natural $k$ and all real $h,t_1,\dots,t_k$, the joint distribution of $(X_{t_1+h},\dots,X_{t_k+h})$ is the $k$-variate normal distribution with the mean vector $(m,\dots,m)\in\Bbb R^k$ and covariance matrix $(g((t_i+h)-(t_j+h))_{i,j=1}^k=(g(t_i-t_j))_{i,j=1}^k$. So, the joint distribution of $(X_{t_1+h},\dots,X_{t_k+h})$ does not depend on $h$; that is, the process $(X_t)_{t\in\Bbb R}$ is stationary (in the narrow sense).