Continuity in the Hurst index of the local times of anisotropic Gaussian random fields

Stochastic Processes and their Applications

Volumn 119, Issue 6, 2009, Pages 1823-1844

  Wu, Dongsheng ^a   Xiao, Yimin ^b  

^a University of Alabama in Huntsville ^*  (United States)

^b MICHIGAN STATE UNIVERSITY   (United States)

Author keywords

Convergence in law; Gaussian random fields; Local times; Sectorial local nondeterminism

Indexed keywords

CONTINUOUS FUNCTIONS; CONVERGENCE IN LAW; FRACTIONAL BROWNIAN MOTIONS; GAUSSIAN RANDOM FIELDS; HURST INDICES; HURST PARAMETERS; LOCAL TIME; LOCAL TIMES; SECTORIAL LOCAL NONDETERMINISM;

ANISOTROPY; BROWNIAN MOVEMENT; LAWS AND LEGISLATION; ROUTERS; TRELLIS CODES;

GAUSSIAN DISTRIBUTION;

EID: 64549090713     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2008.09.001     Document Type: Article

References (33)

1. Davies S., and Hall P. Fractal analysis of surface roughness by using spatial data. J. Roy. Statist. Soc. Ser. B 61 (1999) 3-37 (with discussion)
2. Christakos M.J. Modern Spatiotemporal Geostatistics (2000), Oxford University Press
3. Bonami A., and Estrade A. Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl. 9 (2003) 215-236
4. Benson D.A., Meerschaert M.M., Baeumer B., and Scheffler H.P. Aquifer operator-scaling and the effect on solute mixing and dispersion. Water Res. Res. 42 1 (2006) W01415 18 pp
5. Kamont A. On the fractional anisotropic Wiener field. Probab. Math. Statist. 16 (1996) 85-98
6. Benassi A., Jaffard S., and Roux D. Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13 (1997) 19-90
7. Dalang R.C. Extending martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e's. Electron. J. Probab. 4 6 (1999) 1-29
8. Øksendal B., and Zhang T. Multiparameter fractional Brownian motion and quasi-linear stochastic partial differential equations. Stochast. Stochast. Rep. 71 (2000) 141-163
9. Mueller C., and Tribe R. Hitting probabilities of a random string. Electron. J. Probab. 7 10 (2002) 1-29
10. D. Nualart, Stochastic heat equation driven by fractional noise, 2006, a preprint
11. Eisenbaum N., and Khoshnevisan D. On the most visited sites of symmetric Markov processes. Stochastic Process. Appl. 101 (2002) 241-256
12. Gneiting T. Nonseparable, stationary covariance functions for space-time data. J. Amer. Statist. Assoc. 97 (2002) 590-600
13. Stein M.L. Space-time covariance functions. J. Amer. Statist. Assoc. 100 (2005) 310-321
14. Jolis M., and Viles N. Continuity in law with respect to the Hurst parameter of the local time of the fractional Brownian motion. J. Theoret. Probab. 20 (2007) 133-152
15. Xiao Y. Strong local nondeterminism and sample path properties of Gaussian random fields. In: Lai T.L., Shao Q., and Qian L. (Eds). Asymptotic Theory in Probability and Statistics with Applications (2007), Higher Education Press, Beijing 136-176
16. Y. Xiao, Sample path properties of anisotropic Gaussian random fields, in: D. Khoshnevisan, F. Rassoul-Agha (Eds.), A Minicourse on Stochastic Partial Differential Equations, Springer, New York, 2007 (in press)
17. Wu D., and Xiao Y. Geometric properties of fractional Brownian sheets. J. Fourier Anal. Appl. 13 (2007) 1-37
18. Xiao Y., and Zhang T. Local times of fractional Brownian sheets. Probab. Theory Related Fields 124 (2002) 204-226
19. Ayache A., Wu D., and Xiao Y. Joint continuity of the local times of fractional Brownian sheets. Ann. Inst. H. Poincaré Probab. Statat. 44 (2008) 727-748
20. Ayache A., and Xiao Y. Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets. J. Fourier Anal. Appl. 11 (2005) 407-439
21. Yaglom A.M. Some classes of random fields in n-dimensional space, related to stationary random processes. Theory Probab. Appl. 2 (1957) 273-320
22. Anh V.V., Angulo J.M., and Ruiz-Medina M.D. Possible long-range dependence in fractional random fields. J. Statist. Plann. Inference 80 (1999) 95-110
23. Kahane J.-P. Some Random Series of Functions. 2nd ed. (1985), Cambridge University Press, Cambridge
24. Pitt L.D. Local times for Gaussian vector fields. Indiana Univ. Math. J. 27 (1978) 309-330
25. Robeva R.S., and Pitt L.D. On the equality of sharp and germ σ-fields for Gaussian processes and fields. Pliska Stud. Math. Bulgar. 16 (2004) 183-205
26. Biermé H., Meerschaert M.M., and Scheffler H.-P. Operator scaling stable random fields. Stochastic Process. Appl. 117 (2007) 312-332
27. Geman D., and Horowitz J. Occupation densities. Ann. Probab. 8 (1980) 1-67
28. Dozzi M. Occupation density and sample path properties of N-parameter processes. Topics in Spatial Stochastic Processes (Martina Franca, 2001). Lect. Notes Math. vol. 1802 (2002), Springer, Berlin 127-166
29. Adler R.J. The Geometry of Random Fields (1981), Wiley, New York
30. Khoshnevisan D. Multiparameter Processes: An Introduction to Random Fields (2002), Springer, New York
31. Kallenberg O. Foundations of Modern Probability (2002), Springer, New York
32. Cuzick J., and DuPreez J. Joint continuity of Gaussian local times. Ann. Probab. 10 (1982) 810-817
33. Khoshnevisan D., and Xiao Y. Images of the Brownian sheet. Trans. Amer. Math. Soc. 359 (2007) 3125-3151