Can someone tell me why Cov(X(t), X(t+S) - X(t)) = 0? (pg. 31 - Course notes). I figure it has something to do with independence but I'm not quite sure... Thanks.
Yes, you're right, it is to do with independence. X(t+s) - X(t) is the increment of the process between times t and t+s (ie the number of events occurring in that time.) And this amount is completely independent of X(t) (which is the number of events occuring up to time t). Finally the covariance of two independent variables is zero. That make sense?
