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Higher-Order Allan Variance for Atomic Clocks of Arbitrary Order: Mathematical Foundation

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Takayuki Ishizaki, Taichi Ichimura, Takahiro Kawaguchi, Yuichiro Yano, Yuko Hanado

Probability · Submitted 11 Feb 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv ID2302.05556 (version 3)
Pith IDC6MNRKNZ, Pith's permanent record of this paper and its review
CitationTaichi Ichimura, Takahiro Kawaguchi, Takayuki Ishizaki, Yuichiro Yano, Yuko Hanado. "Higher-Order Allan Variance for Atomic Clocks of Arbitrary Order: Mathematical Foundation." arXiv:2302.05556 (2023). https://pith.science/paper/C6MNRKNZ
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,973 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.24 · compares with prior results 0.04 · mentions code or data availability 0.02.

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arXiv listing abstract v3. Results and the main claim are underlined; .

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.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this paper, we perform a time-domain analysis of the higher-order Allan variance for atomic clock models of arbitrary order. contribution 0.97 0.95 0.03 asserted 0.93
s1 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. method 0.95 0.71 0.03 asserted 1.00
s2 The main results of this paper are threefold. contribution 0.86 0.56 0.04 asserted 0.60
s3 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. result 0.98 0.94 0.23 asserted 1.00
s4 Second, we derive a complete analytical expression of the higher-order Allan variance, which consists of both time-dependent and time-independent terms. result 0.96 0.97 0.15 asserted 1.00
s5 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. result 0.99 0.95 0.32 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

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.

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