Why can correlograms indicate non-stationarity? I'm reading about correlograms, and how they can be used to detect non-stationarity.
Supposedly, if the autocorrelation constant is significant, and/or declines slowly, we would deem the time series non-stationary.
An example of this can be seen here :
http://3.bp.blogspot.com/-4aVTQ9XbMRc/T8lNSFLc8FI/AAAAAAAAAgg/C2otWD3QInw/s1600/corr_inc1.gif 
(source: davegiles.blogspot.com)
But why is this the case?
The definition of stationarity does not state that there can be no covariance/autocorrelation. It only states that the covariance needs to be the same between say, $Y_t$ and $Y_{t+1}$ as between $Y_{t+1}$ and $Y_{t+2}$)
Shouldn't we need "multiple" correlograms, to check if this were the case?
That is, to check that the covariance at the same number of lags is the same, even with different starting points?
I'm taking my first year of statistics, so please do say if I've gotten something wrong ^^
 A: The quote in your comment claims too much but does relate to something real, and that something can be useful in figuring out suitable models for data.
If you have an $I(1)$ series (a very specific kind of nonstationarity), you should see an ACF that doesn't exhibit the kind of geometric "decay" in the characteristic manner that you see with data generated by typical lowish-order stationary ARMA. [There will still be a tendency to decrease in the ACF of an $I(1)$ series, but it often tends to look more "linear" than geometric]
[You can get some sense of this by actually generating data from stationary low order AR and ARMA models and comparing the ACFs from (say) that of a random walk. It's worth trying for a number of different models]
You will also tend to see it with $I(2)$ (etc) series, but data that's reasonably modelled by $I(1)$ or is at least stationary after differencing tends to be more common.
So if you regard the major possibilities as either the series being I(1) or low-to-moderate order stationary ARMA, then the ACF can sometimes be of some help in distinguishing them (but you'd typically difference and look again before saying too much).
