I'm new to stachastic processes and unsure about stationarity. In my script it says a weak stationary gaussian process is always a strictly stationary process as well. I have an example of a weak stationary gaussian process that isn't strictly stationary in my head and don't see the fault.
Lets say we have a an index set of natural numbers $i$ from $1$ to $10$. For each index we have a normal distributed random variable $Z_i \ ( i = 1,...,10)$ with mean $0$ and variance $i$ so that our first random variable is $N(0,1)$ distributed, the second one ist $N(0,2)$ distributed and so on, till the 10th that is $N(0,10)$. Further we say, they are all independent, so the covariancefunction is constantly equal to zero. Then we have a gaussian process that's weak stationary, right? But it's not strictly stationary because the $Z_i$ are obviously not equally distributed.
What am I thinking wrong? Thank you.
The covariance function $C(i, j)$ also includes information about the variance $C(i,i) = \text{Var}(Z_i)$, so weak stationarity implies the variance is constant over time as well.