Asymptotic properties of integral functionals of fractional Brownian fields
Author:
V. I. Makogin
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 91 (2015), 105-114
MSC (2010):
Primary 60J55, 60G60; Secondary 60G18
DOI:
https://doi.org/10.1090/tpms/970
Published electronically:
February 4, 2016
MathSciNet review:
3364127
Full-text PDF Free Access
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Additional Information
Abstract: Two theorems describing the asymptotic behavior of integral functionals of multidimensional self-similar random fields are proved. For a $d$-dimensional fractional Brownian field depending on $N$ parameters, a theorem on the convergence of the integral mean-type functional is established. The weak convergence of an integral functional of a $d$-dimensional anisotropic self-similar random field with $N$ parameters to the local time is proved under the assumption that the continuous local time exists for this field.
References
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References
- A. Ayache, D. Wu, and Y. Xiao, Joint continuity of the local times of fractional Brownian sheets, Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), no. 4, 727–748. MR 2446295 (2010b:60112)
- D. Benson, M. M. Meerschaert, B. Bäumer, and H.-P. Scheffler, Aquifer operator scaling and the effect on solute mixing and dispersion, Water Resour. Res. 42 (2006), 1–18.
- A. Bonami and A. Estrade, Anisotropic analysis of some Gaussian models, J. Fourier Anal. Appl. 9 (2003), 215–236. MR 1988750 (2004e:60082)
- S. Davies and P. Hall, Fractal analysis of surface roughness by using spatial data (with discussion), J. Roy. Statist. Soc. Ser. B 61 (1999), 3–37. MR 1664088 (99i:62124)
- D. Geman and J. Horowitz, Occupation densities, Ann. Probab. 8 (1980), 1–67. MR 556414 (81b:60076)
- G. Kallianpur and H. Robbins, Ergodic property of the Brownian motion process, Proc. Nat. Acad. Sci. 39 (1953), 525–533. MR 0056233 (15:44g)
- N. Kôno, Kallianpur–Robbins law for fractional Brownian motion, Probability Theory and Mathematical Statistics, Proc. 7th Japan–Russia Symp., 1996, pp. 229–236. MR 1467943 (98e:60058)
- W. Z. Lu and X. K. Wang, Evolving trend and self-similarity of ozone pollution in central Hong Kong ambient during 1984–2002, Sci. Total Environ. 357 (2006), 160–168.
- V. I. Makogin and Yu. S. Mishura, Strong limit theorems for anisotropic self-similar fields, Modern Stoch. Theory Appl. 1 (2014), 73–94 MR 3314795
- A. Morata, M. L. Martin, M. Y. Luna, and F. Valero, Self-similarity patterns of precipitation in the Iberian Peninsula, Theor. Appl. Climatol. 85 (2006), 41–59.
- N.-R. Shieh, Limit theorems for local times of fractional Brownian motions and some other self-similar processes, J. Math. Kyoto Univ. 36 (1996), 641–652. MR 1443740 (98f:60077)
- Y. Xiao and T. Zhang, Local times of fractional Brownian sheets, Probab. Theory Related Fields 124 (2002), no. 2, 204–226. MR 1936017 (2004b:60108)
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Additional Information
V. I. Makogin
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
makoginv@ukr.net
Keywords:
Local time,
self-similar fields,
anisotropic fractional Brownian field
Received by editor(s):
September 30, 2014
Published electronically:
February 4, 2016
Article copyright:
© Copyright 2016
American Mathematical Society
