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OPERATOR FRACTIONAL BROWNIAN SHEET AND MARTINGALE DIFFERENCES

  • Dai, Hongshuai (School of Statistics Shandong University of Finance and Economics) ;
  • Shen, Guangjun (Department of Mathematics Anhui Normal University) ;
  • Xia, Liangwen (Department of Mathematics Anhui Normal University)
  • Received : 2016.08.25
  • Accepted : 2017.10.17
  • Published : 2018.01.31

Abstract

In this paper, inspired by the fractional Brownian sheet of Riemann-Liouville type, we introduce the operator fractional Brownian sheet of Riemman-Liouville type, and study some properties of it. We also present an approximation in law to it based on the martingale differences.

Acknowledgement

Supported by : National Natural Science Foundation of China, Shandong Natural Science Foundation, Distinguished Young scholars Foundation of Anhui Province, University Discipline

References

  1. A. Ayache, S. Leger, and M. Pontier, Drap brownien fractionnaire, Potential Anal. 17 (2002), no. 1, 31-43. https://doi.org/10.1023/A:1015260803576
  2. X. Bardina and M. Jolis, Weak approximation of the Brownian sheet from a Poisson process in the plane, Bernoulli 6 (2000), no. 4, 653-665. https://doi.org/10.2307/3318512
  3. X. Bardina, M. Jolis, and C. A. Tudor, Weak convergence to the fractional Brownian sheet and other two-parameter Gaussian processes, Statist. Probab. Lett. 65 (2003), no. 4, 317-329. https://doi.org/10.1016/j.spl.2003.09.001
  4. J. A. Barnes and D. W. Allan, A statistical model of flicker noise, Proc. IEEE. 54 (1966), 176-178. https://doi.org/10.1109/PROC.1966.4630
  5. P. J. Bickel and M. J. Wichura, Convergence criteria for multiparameter stochastic processes and some applications, Ann. Math. Statist. 42 (1971), 1656-1670. https://doi.org/10.1214/aoms/1177693164
  6. R. Cairoli and J. B. Walsh, Stochastic integrals in the plane, Acta Math. 134 (1975), 111-183. https://doi.org/10.1007/BF02392100
  7. H. Dai, Convergence in law to operator fractional Brownian motion of Riemann-Liouville type, Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 4, 777-788.
  8. H. Dai, Convergence in law to operator fractional Brownian motions, J. Theoret. Probab. 26 (2013), no. 3, 676-696. https://doi.org/10.1007/s10959-011-0401-4
  9. H. Dai, Approximation to multifractional Riemann-Liouville Brownian sheet, Comm. Statist. Theory Methods 44 (2015), no. 7, 1399-1410. https://doi.org/10.1080/03610926.2013.765476
  10. H. Dai, T.-C. Hu, and J.-Y. Lee, Operator fractional Brownian motion and martingale differences, Abstr. Appl. Anal. 2014, Art. ID 791537, 8 pp.
  11. H. Dai and Y. Li, A note on approximation to multifractional Brownian motion, Sci. China Math. 54 (2011), no. 10, 2145-2154. https://doi.org/10.1007/s11425-011-4246-1
  12. H. Dai, G. Shen, and L. Kong, Limit theorems for functionals of Gaussian vectors, Front. Math. China 12 (2017), no. 4, 821-842. https://doi.org/10.1007/s11464-016-0620-1
  13. G. Didier and V. Pipiras, Integral representations and properties of operator fractional Brownian motions, Bernoulli 17 (2011), no. 1, 1-33. https://doi.org/10.3150/10-BEJ259
  14. G. Didier and V. Pipiras, Exponents, symmetry groups and classification of operator fractional Brownian motions, J. Theoret. Probab. 25 (2012), no. 2, 353-395. https://doi.org/10.1007/s10959-011-0348-5
  15. A. M. Garsia, Continuity properties of Gaussian processes with multidimensional time parameter, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statis-tics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, 369-374, Univ. California Press, Berkeley, CA.
  16. A. Kamont, On the fractional anisotropic Wiener field, Probab. Math. Statist. 16 (1996), no. 1, 85-98.
  17. R. G. Laha and V. K. Rohatgi, Operator self-similar stochastic processes in Rd, Sto-chastic Process. Appl. 12 (1982), no. 1, 73-84.
  18. J. Lamperti, Semi-stable stochastic processes, Trans. Amer. Math. Soc. 104 (1962), 62-78. https://doi.org/10.1090/S0002-9947-1962-0138128-7
  19. Y. Li and H. Dai, Approximations of fractional Brownian motion, Bernoulli 17 (2011), no. 4, 1195-1216. https://doi.org/10.3150/10-BEJ319
  20. S. C. Lim, Fractional Brownian motion and multifractional Brownian motion of Riemann-Liouville type, J. Phys. A 34 (2001), no. 7, 1301-1310. https://doi.org/10.1088/0305-4470/34/7/306
  21. M. Maejima and J. D. Mason, Operator-self-similar stable processes, Stochastic Process. Appl. 54 (1994), no. 1, 139-163. https://doi.org/10.1016/0304-4149(94)00010-7
  22. J. D. Mason and Y. Xiao, Sample path properties of operator-self-similar Gaussian random fields, Theory Probab. Appl. 46 (2002), no. 1, 58-78. https://doi.org/10.1137/S0040585X97978749
  23. R. Morkvenas, The invariance principle for martingales in the plane, Litovsk. Mat. Sb. 24 (1984), no. 4, 127-132.
  24. G. Samorodnitsky and M. S. Taqqu, Stable non-Gaussian random processes, Stochastic Modeling, Chapman & Hall, New York, 1994.
  25. W. Vervaat, Sample path properties of self-similar processes with stationary increments, Ann. Probab. 13 (1985), no. 1, 1-27. https://doi.org/10.1214/aop/1176993063
  26. Z. Wang, L. Yan, and X. Yu, Weak approximation of the fractional Brownian sheet from random walks, Electron. Commun. Probab. 18 (2013), no. 90, 13 pp.
  27. Z. Wang, L. Yan, and X. Yu, Weak convergence to the fractional Brownian sheet using martingale differ-ences, Statist. Probab. Lett. 92 (2014), 72-78. https://doi.org/10.1016/j.spl.2014.04.014
  28. W. Whitt, Stochastic-process limits, Springer Series in Operations Research, Springer-Verlag, New York, 2002.