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?
$\endgroup$1 Answer
$\begingroup$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.
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