When is a Gaussian Process Stationary? Recall that a stochastic process is stationary if its finite-dimensional distributions are invariant under shifts in time. This is sometimes called strong...

When is a Gaussian Process Stationary? Recall that a stochastic process is stationary if its finite-dimensional distributions are invariant under shifts in time. This is sometimes called strong stationarity. A related notion is that a real-valued process {X(t) : t ≥ 0} is weakly stationary if its mean function E[X(t)] is a constant and its covariance function Cov(X(s), X(t)) depends only on |t−s|. Weak stationarity does not imply strong stationarity. However, if a real-valued process is strongly stationary and its mean and covariance functions are finite, then the process is weakly stationary. Show that a Gaussian process is strongly stationary if and only if it is weakly stationary.