My understanding re. ADEV is that it calculates the standard deviation of the oscillator's signal over different time intervals and plots those deviations as a function of the time interval.

No, it's not calculating standard deviations.

One reason Allan came up with the Allan Variation is the traditional standard deviation of real oscillators diverges as the sample size increases.

I began thinking about how to convert an Allan Variance/Deviation into a probabilistic bounds on frequency during a particular interval. However, it quickly became apparent that this is not a simple problem.

The Allan Variance uses the average frequency, f

_{i} (measured in radians/sec), over an averaging interval tau. These are normalized by the nominal frequency to ensure the Allan Variances of oscillators with different frequencies, w

_{0}, are comparable. This normalization produces what is called fractional frequency data ff

_{i} = f

_{i}/w

_{0}. Suppose these samples are generated by a stationary process. Then the standard variance is simple to compute.

However, the Allan Variance is a function of the differenced fractional frequency data: a

_{i} = ff

_{i+1}-ff

_{i}. It sums the square of these values and averages the sum (dividing by 2). The time series a

_{i} is autocorrelated. Now, it is possible for a stationary process to produce an autocorrelated series, but this is generally not the case. So, it is possible (likely) that a

_{i} represents samples from a non-stationary process. (Someone who is more knowledgable than I can correct me on this.) If so, the Allan Variance will not have the same properties as a standard variance. In particular, you can't use the Allan Deviation as you would a standard deviation from some pdf, defining probabilistic bounds based on it.

However, I am not an expert on Allan Variance/Deviation, so maybe there is some way to use it to compute the desired bounds. This is what I have been asking for someone to explain in recent posts.