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.

翻译：本文对任意阶原子钟模型的高阶Allan方差进行了时域分析。采用标准原子钟模型，其中钟读数偏差的时间序列表示为Wiener过程或积分Wiener过程，我们将高阶Allan方差定义为钟读数偏差的高阶差分的均方值。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操作（可解释为具有二项式系数的线性聚合）不仅是得到独立同分布高斯过程的充分条件，也是必要条件。其次，推导了高阶Allan方差的完整解析表达式，该表达式包含时间依赖项和时间独立项。最后，证明了当且仅当差分操作的阶数大于或等于原子钟模型的阶数时，高阶Allan方差与时间无关。