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.

翻译：本文分析了任意阶原子钟模型的高阶阿伦方差。采用标准原子钟模型，其中钟读数偏差的时间序列表示为维纳过程或积分维纳过程，我们将高阶阿伦方差定义为钟读数偏差的高阶差分的均方值。本文的主要成果有三点。首先，我们证明钟读数偏差的高阶差分运算（可解释为二项式系数的线性聚合）不仅充分，而且必要，能使聚合后的时间序列成为独立同分布的高斯过程。其次，我们推导出高阶阿伦方差的完整解析表达式，包含时间相关项和时间无关项。最后，我们证明高阶阿伦方差与时间无关当且仅当差分阶数大于或等于原子钟模型的阶数。