What is the difference between a weakly stationary process and strictly stationary process?

Question

In some lecture slides I read that the definition of a weakly stationary process is that

• The mean value is constant
• The covariance function is time-invariant
• The variance is constant

and I read that the definition of a strictly stationary process is a process whose probability distribution does not change over time.

What concrete properties of a strictly stationary process is not included in the definition of a weakly stationary process?

Answer 1

The information about distributions is not included. Say, you may consider the trivial example of a discrete time process $X_n$, $n=1,2,\ldots$ with independent values s.t.:

$X_1$ is Poisson distributed with mean $1$ and variance $1$,

$X_2$ is exponentially distributed with mean $1$ and variance $1$,

$X_3$ takes values $0$ and $2$ with equal probabilities and also has mean $1$ and variance $1$,

$X_4$ is normally distributed with mean $1$ and variance $1$

and so on

Then the mean value is a constant, the variance is a constant, covariances are zero and the probability distribution depends on a time.