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Acta Mathematica Vietnamica

, Volume 42, Issue 4, pp 637–651 | Cite as

Quantitative Central Limit Theorems of Spherical Sojourn Times of Isotropic Gaussian Fields

  • Pham Viet HungEmail author
Article
  • 71 Downloads

Abstract

We investigate the rate of convergence for the central limit theorems of sojourn times on the growing sphere of isotropic Gaussian fields defined on the whole Euclidean space. In the case of the sojourn times defined on a cube, the similar problem has been studied by using the Malliavin-Stein method. Following this idea, in this paper, we establish the explicit rate for various probability distances with a careful examination of the variance.

Keywords

Sojourn time Isotropic Gaussian field Malliavin-Stein method 

Mathematics Subject Classification (2010)

Primary 60H07 Secondary 60F05 60D05 

Notes

Acknowledgements

This work is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.03-2014.14. We would like to thank the anonymous referee for his meticulous and rigorous reading of the manuscript and his numerous suggestions that greatly improve the presentation of the present paper.

References

  1. 1.
    Arcones, M.A.: Limit theorems for nonlinear functionals of a stationary gaussian sequence of vectors. Ann. Probab. 22(4), 2242–2274 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berman, S.M.: Sojourns and extremes of stochastic processes. The Wadsworth & Brooks/Cole Statistics/Probability Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA (1992)CrossRefzbMATHGoogle Scholar
  3. 3.
    Biermé, H., Bonami, A., León, J. R.: Central limit theorems and quadratic variations in terms of spectral density. Electron. J. Probab. 16(13), 362–395 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Breuer, P., Major, P.: Central limit theorems for nonlinear functionals of gaussian fields. J. Multivar. Anal. 13(3), 425–441 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bulinski, A., Spodarev, E., Timmermann, F.: Central limit theorems for the excursion set volumes of weakly dependent random fields. Bernoulli 18(1), 100–118 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cammarota, V., Marinucci, D.: On the limiting behaviour of needlets polyspectra. Ann. Inst. Henri Poincaré Probab. Stat. 51(3), 1159–1189 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chambers, D., Slud, E.: Central limit theorems for nonlinear functionals of stationary gaussian processes. Probab. Theory Relat. Fields 80(3), 323–346 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, L.H.Y.: Stein meets malliavin in normal approximation. Acta Math. Vietnam. 40(2), 205–230 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Giraitis, L., Surgailis, D.: Clt and other limit theorems for functionals of gaussian processes. Z. Wahrsch. Verw. Gebiete 70(2), 191–212 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ivanov, A.V., Leonenko, N.N.: Statistical analysis of random fields. Volume 28 of Mathe- matics and its applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1989)Google Scholar
  11. 11.
    Kim, Y.T., Park, H.S.: Kolmogorov distance for the central limit theorems of the wiener chaos expansion and applications. J. Korean Statist. Soc. 44(4), 565–576 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Leonenko, N.: Limit theorems for random fields with singular spectrum. Volume 465 of mathematics and its applications. Kluwer Academic Publishers, Dordrecht (1999)CrossRefGoogle Scholar
  13. 13.
    Leonenko, N., Olenko, A.: Sojourn measures of student and fisher-snedecor random fields. Bernoulli 20(3), 1454–1483 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Marinucci, D., Rossi, M.: Stein-malliavin approximations for nonlinear functionals of random eigenfunctions on \(\mathbb {S}^{d}\). J. Funct. Anal. 268(8), 2379–2420 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Marinucci, D., Wigman, I.: On the area of excursion sets of spherical Gaussian eigenfunctions. J. Math. Phys. 52(9), 21 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Marinucci, D., Wigman, I.: On nonlinear functionals of random spherical eigenfunctions. Comm. Math. Phys. 327(3), 849–872 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nourdin, I., Peccati, G.: Stein’s method on wiener chaos. Probab. Theory Relat. Fields 145(1-2), 75–118 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nourdin, I., Peccati, G., Podolskij, M.: Quantitative Breuer-Major theorems. Stoch. Process. Appl. 121(4), 793–812 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nualart, D.: The Malliavin calculus and related topics, 2nd edn. Probability and its applications (New York). Springer-Verlag, Berlin (2006)zbMATHGoogle Scholar
  20. 20.
    Nualart, D., Peccati, G.: Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33(1), 77–193 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pham, V.H.: On the rate of convergence for central limit theorems of sojourn times of gaussian fields. Stoch. Process. Appl. 123(6), 2158–2174 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2017

Authors and Affiliations

  1. 1.Institute of Mathematics, Vietnam Academy of Science and Technology (VAST)HanoiVietnam

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