Higher-Order Allan Variance for Atomic Clocks of Arbitrary Order

In this paper, we analyze the higher-order Allan variance for atomic clock models of arbitrary order. Adopting a standard atomic clock model where the time series of the clock reading deviation is expressed as a Wiener or integrated Wiener process, we define the higher-order Allan variance as the mean squared higher-order difference of clock reading deviation. The main results of this paper are threefold. First, we prove that the higher-order difference operation of clock reading deviation, which can be interpreted as a linear aggregation with binomial coefficients, is not only sufficient, but also necessary for a resultant aggregated time series to be an independent and identically distributed Gaussian process. Second, we derive a complete analytical expression of the higher-order Allan variance, composed of both time-dependent and time-independent terms. Third and finally, we prove that the higher-order Allan variance is time independent if and only if the order of difference is greater than or equal to the order of atomic clock models.