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

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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.

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Data sharing is not applicable to this article, since not data sets were generated or analyzed during the study.

References

  1. Aitchison, J.: The Statistical Analysis of Compositional Data. Chapman and Hall Ltd, London (1986)

    Book  MATH  Google Scholar 

  2. Anderes, E., Moller, J., Rasmussen, J.G.: Isotropic covariance functions on graphs and their edges. Ann. Statist. 48, 2478–2503 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge, UK (1999)

    Book  MATH  Google Scholar 

  4. Askey, R., Bingham, N.H.: Gaussian processes on compact symmetric spaces. Z. Wahrscheinlichkeit- stheorie und Verw. Gebiete 37, 127–143 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barndorff-Nielsen, O.E.: Some parametric models on the simplex. J. Mul. Anal. 39, 106–116 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions. Springer, New York (1984)

    Book  MATH  Google Scholar 

  7. Bingham, N.H., Mijatovic, A., Symons, T.L.: Brownian manifolds, negative type and geotemporal covariances. Com. Stoch. Anal. 10, 421–432 (2016)

    Google Scholar 

  8. Bos, L., Levenberg, N., Waldron, N.: Metrics associated to multivariate polynomial inequalities. In: Advances in Constructive Approximation (Vanderbilt 2003), pp. 133-147. Brentwood, TN: Nashboro Press (2004)

  9. Cheng, D., Xiao, Y.: Excursion probability of Gaussian random fields on sphere. Bernoulli 22, 1113–1130 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cohen, S., Lifshits, M.A.: Stationary Gaussian random fields on a hyperbolic space and on Euclidean spheres. ESAIM: Prob. Statist 16, 165–221 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dai, F., Xu, Y.: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer, New York (2013)

    Book  MATH  Google Scholar 

  12. Fang, K., Kotz, S., Ng, K.: Symmetric Multivariate and Related Distributions. Chapman and Hall, London (1990)

    Book  MATH  Google Scholar 

  13. Feng, S.: The Poisson-Dirichlet Distribution and Related Topics: Models and Asymptotic Behaviors. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  14. Gangolli, R.: Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters. Ann Inst H Poincaré, B 3, 121–226 (1967)

    MathSciNet  MATH  Google Scholar 

  15. Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products, 7th edn. Academic Press, Amsterdam (2007)

    MATH  Google Scholar 

  16. Hannan, E.J.: Multiple Time Series. Wiley, New York (1970)

    Book  MATH  Google Scholar 

  17. Istas, J.: Karhunen-Loéve expansion of spherical fractional Brownian motions. Statist. Probab. Lett. 76, 1578–1583 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kaishev, V.K.: Lévy processes induced by Dirichlet (B)-splines: Modeling multivariate asset price dynamics. Math. Finance 23, 217–247 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Karlin, S., Micchelli, C.A., Rinot, Y.: Multivariate splines: A probabilistic perspective. J. Mult. Anal. 20, 69–90 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kass, R.E.: The geometry of asymptotic inference. Statist. Sci. 4, 188–234 (1989)

    MathSciNet  MATH  Google Scholar 

  21. Lan, X., Xiao, Y.: Strong local nondeterminism of spherical fractional Brownian motion. Statist. Probab. Lett. 135, 44–50 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lifshits, M., Volkova, K.: Bifractional Brownian motion: existence and border cases. ESAIM: Probab. Statist. 19, 766–781 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lu, T., Ma, C.: Isotropic covariance matrix functions on compact two-point homogeneous spaces. J. Theor. Probab. 33, 1630–1656 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lu, T., Ma, C., Wang, F.: Series expansions of fractional Brownian motions and strong local nondeterminism of bifractional Brownian motions on balls and spheres. Theor. Probab. Appl. 25, 175004 (2017)

    Google Scholar 

  25. Lu, T., Leonenko, N., Ma, C.: Series representations of isotropic vector random fields on balls. Statist. Probab. Lett. 156, 108583 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ma, C.: Vector random fields with second-order moments or second-order increments. Stoch. Anal. Appl. 29, 197–215 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ma, C.: Covariance matrices for second-order vector random fields in space and time. IEEE Trans. Signal Proc. 59, 2160–2168 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ma, C.: Multifractional vector Brownian motions, their decompositions, and generalizations. Stoch. Anal. Appl. 33, 535–548 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ma, C.: Time varying isotropic vector random fields on spheres. J. Theor. Probab. 30, 1763–1785 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ma, C., Malyarenko, A.: Time-varying isotropic vector random fields on compact two-point homogeneous spaces. J. Theor. Prob. 33, 319–339 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Malyarenko, A.: Invariant Random Fields on Spaces with a Group Action. Springer, New York (2013)

    Book  MATH  Google Scholar 

  32. McNeil, A.J., Neslehová, J.: Multivariate Archimedean copulas, d-monotone functions and \(\ell _1\)-norm symmetric distributions. Anna. Statist. 37, 3059–3097 (2009)

    MATH  Google Scholar 

  33. Szegö, G.: Orthogonal Polynomials, AMS Coll. Publ., Vol. 23. Providence, RI (1975)

  34. Talarczyk, A.: Bifractional Brownian motion for \(H > 1\) and \( 2 H K \le 1\). Statist. Probab. Lett. 157, 108628 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  35. Venet, N.: Nonexistence of fractional Brownian fields indexed by cylinders. Electron. J. Probab. 24, 1–26 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, F., Ma, C.: \(\ell _1\)-symmetric vector random fields. Stoch. Proc. Appl. 129, 2466–2484 (2019)

    Article  MATH  Google Scholar 

  37. Yaglom, A.M.: Correlation Theory of Stationary and Related Random Functions, vol. I. Springer, New York (1987)

    Book  MATH  Google Scholar 

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Acknowledgements

The author would like to thank an anonymous referee very much for careful reading and useful comments, which lead to a better presentation and to an improved readability of this work.

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  1. Department of Mathematics, Statistics and Physics, Wichita State University, Wichita, KS, 67260-0033, USA

    Chunsheng Ma

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  1. Chunsheng Ma
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Correspondence to Chunsheng Ma.

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Ma, C. Vector Random Fields on the Probability Simplex with Metric-Dependent Covariance Matrix Functions. J Theor Probab 36, 1922–1938 (2023). https://doi.org/10.1007/s10959-022-01217-6

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