I read it in various references that the gaussian process will be also SSS ( Strict - Sense Stationary) if it is WSS because the pdf is just defined by the covariance matrix and the mean vector.
My main confusion is that, I agree there will no dependency on time of the nth order PDF caused by the mean vector but isn't the covariance matrix still dependent on time , ($t_{1} - t_{2}$ ).
To the extent to what I understand( which might be wrong), the dependency on time of the covariance matrix in the nth order PDF of a GP only eliminate if we assume that all the $t_{i}$ are consecutive. So what is the actual explanation ?
Thanks For help.
I think you might be missing the meaning of strict sense stationarity, which is that for any number of finite times, if you shift all those times by the same amount, the distribution is still the same. Thus, your strict sense stationarity definition also relies on the differences of times in a certain sense of the word
So, if the density $f$ only depends on $t_1 - t_2$, then since $t_1 - t_2 = (t_1 + \tau) - (t_2 + \tau)$, then you will have $f(t_1, t_2) = f(t_1 + \tau, t_2 + \tau)$ for any $\tau$.