Why is it that we require covariance to depend only on the time lag also (besides constant mean and variance, which is intuitive to me),for a process to be weakly stationary.
I'm not too sure what to answer That's the way that weak stationarity is defined - in terms of the mean and covariance. When thinking about a stochastic process (ie a sequence of random variables), we think about the covariance function as this gives us information about the way different values in the sequence relate to each other, eg how is X5 related to X8. Because stationarity in general is to do with the statistical properties of the process not changing over time, we want the dependence (or covariance) of X5 and X8, to be the same as the dependence of X13 and X16 (for example), as they are both separated by the same time lag. Considering the covariance function includes consideration of the variance (since cov(Xt,Xt) = var(Xt)), and more information besides.
