My current work involves the design details of atomic clocks where we use the Allan Variance and Allan Deviation (ADEV) extensively. The primary point is that it can be used for non-stationary processes (which frequency noise is). For non-stationary signals where the autocorrelation or power spectral density can't be used with consistency, the Allan Deviation will converge to a consistent metric for comparison of the frequency accuracy of two different clocks (and generally for the comparison of the noise in non-stationary signals).
Non-stationary signals will often have a shorter time duration over which the signal can be sufficiently assumed to be stationary. Oscillator phase noise is an excellent example of this: for larger frequency offsets (typically 1 Hz and above for lower frequency oscillators such as 10MHz) a power spectral density (PSD) can be computed and given as the oscillator’s phase noise performance. The issue is this PSD cannot be computed consistently at lower frequency offsets as the non—stationary contributions start to dominate and this is where the ADEV computation will shine: in computing the longer term frequency accuracy of the clock over time durations where non-stationary contributions dominate (1/f noise, drift). ADEV is computed as an rms error given by the difference between the average frequency accuracy over interval tau seconds compared to the average frequency accuracy tau seconds ago. The rms of this error is computed over a very long duration (this error difference is a stationary signal and thus the standard deviation can be consistently computed).
One specific utility for me that the ADEV has provided well beyond the world of clocks is to determine the time constraints under which a random process can be sufficiently assumed to be stationary: if the random process is stationary we can continue to average to get a better estimate of the underlying mean— therefore we can use the ADEV to determine the optimum averaging time after which no further improvement can be gained (and through further averaging our estimate can get worst). This applies to everything from channel estimation to financial markets, wherever we may be observing past results to make an estimate of an underlying process.
I explain further details of the Allan Deviation and its potential use and application well beyond oscillator stability in these other posts:
How to interpret Allan Deviation plot for gyroscope?
What determines the accuracy of the phase result in a DFT bin?
Averaging data from 2 sensors
Is it possible to add some kind of noise to a non stationary (having unit root) signals to make them stationary?