Allan variance of time series models for measurement data

The uncertainty of the mean of autocorrelated measurements from a stationary process has been discussed in the literature. However, when the measurements are from a non-stationary process, how to assess their uncertainty remains unresolved. Allan variance or two-sample variance has been used in time and frequency metrology for more than three decades as a substitute for the classical variance to characterize the stability of clocks or frequency standards when the underlying process is a 1/f noise process. However, its applications are related only to the noise models characterized by the power law of the spectral density. In this paper, from the viewpoint of the time domain, we provide a statistical underpinning of the Allan variance for discrete stationary processes, random walk and long-memory processes such as the fractional difference processes including the noise models usually considered in time and frequency metrology. Results show that the Allan variance is a better measure of the process variation than the classical variance of the random walk and the non-stationary fractional difference processes including the 1/f noise.