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, 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.

翻译：本文针对任意阶原子钟模型的高阶阿伦方差进行了时域分析。采用标准原子钟模型，将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的均方值。本文的主要结果有三方面：首先，我们证明了钟读数偏差的高阶差分运算（可解释为具有二项式系数的线性聚合）不仅是充分条件，也是必要条件，使得聚合后的时间序列成为独立同分布的高斯过程。其次，我们推导了高阶阿伦方差的完整解析表达式，该表达式由时间相关项和时间无关项共同构成。第三，我们证明了高阶阿伦方差与时间无关的充要条件是差分运算的阶数大于或等于原子钟模型的阶数。