Hi, I'm confused about why the notes on p20 of chapter 2 say that white noise does not have independent increments. I don't understand this. Isn't white noise just completely random? So the difference between 2 points doesn't depend on the past? Can anyone help?
If, say, X(1) was a very low value (much less than the mean), and X(2) was very high value (much more than the mean), then the increment is a large positive value. Would you expect the next increment to be large and positive again? No - it would be much more likely for it to be negative. That is, the probability of getting a future increment of any particular size is influenced by the value of the increment you have just had, ie they are not independent. In fact, the size of the next increment is actually dependent on the value of the current value of the state, but because this current value is a function of the past increments of the process, then the increments will also be non-independent of each other. Make sense??
