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Circular autocorrelation of stationary circular Markov processes

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The stationary Markov process is considered and its circular autocorrelation function is investigated. More specifically, the transition density of the stationary Markov circular process is defined by two circular distributions, and we elucidate the structure of the circular autocorrelation when one of these distributions is uniform and the other is arbitrary. The asymptotic properties of the natural estimator of the circular autocorrelation function are derived. Furthermore, we consider the bivariate process of trigonometric functions and provide the explicit form of its spectral density matrix. The validity of the model was assessed by applying it to a series of wind direction data.

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  1. 1.

    The similar term “circular serial correlation coefficient” is used in Anderson (1971, Section 6.5.2) in a different context, where he considered a process whose first and last observations are connected to each other.

  2. 2.

    The cacf function at lag \(k(\ge 0)\) in Holzmann et al. (2006) should be read as \(R_{\mathrm {C}}(k)=(I_1(\kappa )/I_0(\kappa ))^{2k}\). Here, \(I_r(\kappa )\) is the modified Bessel function of the first kind of order r. See Sect. 4 for details.


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The authors would like to express their gratitude to the editor and two anonymous referees, whose invaluable comments improved the paper.

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Correspondence to Hiroaki Ogata.

Additional information

The research reported herein was supported by JSPS KAKENHI Grant Numbers 15K17593, 26380401, 15K21433, and 26870655.

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Abe, T., Ogata, H., Shiohama, T. et al. Circular autocorrelation of stationary circular Markov processes. Stat Inference Stoch Process 20, 275–290 (2017). https://doi.org/10.1007/s11203-016-9154-0

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  • Circular statistics
  • Time series models
  • Toroidal data
  • Wind direction
  • Wrapped cauchy distribution