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When is a truncated covariance function on the line a covariance function on the circle?

Author

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  • Wood, Andrew T. A.

Abstract

Let [gamma] denote the covariance function of a real stationary process on . Define a new function on [-K, K] by [lambda]K(t) = [gamma](t), t [epsilon] [-K, K]. Note that by identifying the end points of the interval, we may interpret [lambda]K as a function on the circle with circumference 2K. We address the following question: if [gamma] is a covariance function on the line, will [lambda]K be a covariance function on the circle? We identify one class of covariance functions for which the answer is "yes" for all K > 0, and a second class for which it is "yes" for all K sufficiently large. However, our most substantial result is a negative one, and the answer will frequently be "no" for all K > 0. A statistical consequence of the positive results is mentioned briefly.

Suggested Citation

  • Wood, Andrew T. A., 1995. "When is a truncated covariance function on the line a covariance function on the circle?," Statistics & Probability Letters, Elsevier, vol. 24(2), pages 157-164, August.
    • Handle: RePEc:eee:stapro:v:24:y:1995:i:2:p:157-164
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    Cited by:

    1. Huang, Chunfeng & Zhang, Haimeng & Robeson, Scott M., 2016. "Intrinsic random functions and universal kriging on the circle," Statistics & Probability Letters, Elsevier, vol. 108(C), pages 33-39.
    2. Huang, Chunfeng & Li, Ao, 2021. "On Lévy’s Brownian motion and white noise space on the circle," Statistics & Probability Letters, Elsevier, vol. 171(C).
    3. Gneiting, Tilmann, 1998. "Simple tests for the validity of correlation function models on the circle," Statistics & Probability Letters, Elsevier, vol. 39(2), pages 119-122, August.
    4. Huang, Chunfeng & Zhang, Haimeng & Robeson, Scott M., 2012. "A simplified representation of the covariance structure of axially symmetric processes on the sphere," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1346-1351.

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