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 ^^