In this paper, we perform a time-domain analysis of 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 the clock reading
deviation. The main results of this paper are threefold. First, we prove that
the higher-order difference operation of the clock reading deviation, which can
be interpreted as a linear aggregation with binomial coefficients, is not only
sufficient, but also necessary for a resulting 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, which
consists 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 operation is greater than or equal to the order of
the atomic clock model