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Vector Random Fields on the Probability Simplex with Metric-Dependent Covariance Matrix Functions

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  • Chunsheng Ma

    (Wichita State University)

Abstract

This paper constructs a class of isotropic vector random fields on the probability simplex via infinite series expansions involving the ultraspherical polynomials, whose covariance matrix functions are functions of the metric (distance function) on the probability simplex, and introduces the scalar and vector fractional, bifractional, and trifractional Brownian motions over the probability simplex, while the metric is shown to be conditionally negative definite.

Suggested Citation

  • Chunsheng Ma, 2023. "Vector Random Fields on the Probability Simplex with Metric-Dependent Covariance Matrix Functions," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1922-1938, September.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:3:d:10.1007_s10959-022-01217-6
    DOI: 10.1007/s10959-022-01217-6
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    References listed on IDEAS

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    1. Barndorff-Nielsen, O. E. & Jørgensen, B., 1991. "Some parametric models on the simplex," Journal of Multivariate Analysis, Elsevier, vol. 39(1), pages 106-116, October.
    2. Lu, Tianshi & Leonenko, Nikolai & Ma, Chunsheng, 2020. "Series representations of isotropic vector random fields on balls," Statistics & Probability Letters, Elsevier, vol. 156(C).
    3. Talarczyk, Anna, 2020. "Bifractional Brownian motion for H>1 and 2HK≤1," Statistics & Probability Letters, Elsevier, vol. 157(C).
    4. Karlin, S. & Micchelli, C. A. & Rinott, Y., 1986. "Multivariate splines: A probabilistic perspective," Journal of Multivariate Analysis, Elsevier, vol. 20(1), pages 69-90, October.
    5. Istas, Jacques, 2006. "Karhunen-Loeve expansion of spherical fractional Brownian motions," Statistics & Probability Letters, Elsevier, vol. 76(14), pages 1578-1583, August.
    6. Chunsheng Ma, 2017. "Time Varying Isotropic Vector Random Fields on Spheres," Journal of Theoretical Probability, Springer, vol. 30(4), pages 1763-1785, December.
    7. Lan, Xiaohong & Xiao, Yimin, 2018. "Strong local nondeterminism of spherical fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 135(C), pages 44-50.
    8. Cheng, Dan, 2016. "Excursion probability of certain non-centered smooth Gaussian random fields," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 883-905.
    9. Tianshi Lu & Chunsheng Ma, 2020. "Isotropic Covariance Matrix Functions on Compact Two-Point Homogeneous Spaces," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1630-1656, September.
    10. Chunsheng Ma & Anatoliy Malyarenko, 2020. "Time-Varying Isotropic Vector Random Fields on Compact Two-Point Homogeneous Spaces," Journal of Theoretical Probability, Springer, vol. 33(1), pages 319-339, March.
    Full references (including those not matched with items on IDEAS)

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