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Properties of Strong Local Nondeterminism and Local Times of Stable Random Fields

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Seminar on Stochastic Analysis, Random Fields and Applications VI

Part of the book series: Progress in Probability ((PRPR,volume 63))

Abstract

We establish properties of strong local nondeterminism for several classes of α-stable random fields such as harmonizable-type fractional stable fields with stationary increments, harmonizable and linear fractional stable sheets. We apply these properties to study existence and joint continuity of the local times of stable random fields.

Mathematics Subject Classification (2000). 60G52; 60G17; 60G18; 60G60.

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References

  1. R.J. Adler, The Geometry of Random Fields, Wiley, New York, 1981.

    Google Scholar 

  2. V.V. Anh, J.M. Angulo, and M.D. Ruiz-Medina, Possible long-range dependence in fractional random fields, J. Statist. Plann. Inference, 80 (1999), 95–110.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Ayache, F. Roueff, and Y. Xiao, Local and asymptotic properties of linear fractional stable sheets, C. R. Acad. Sci. Paris, Ser. A., 344 (2007), 389–394.

    Google Scholar 

  4. A. Ayache, F. Roueff, and Y. Xiao, Joint continuity of the local times of linear fractional stable sheets, C. R. Acad. Sci. Paris, Ser. A., 344 (2007), 635–640.

    Google Scholar 

  5. A. Ayache, F. Roueff, and Y. Xiao, Linear fractional stable sheets: wavelet expansion and sample path properties, Stoch. Process. Appl., 119 (2009), 1168–1197.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Ayache, F. Roueff, and Y. Xiao, Joint continuity and H¨older conditions for the local times of linear fractional stable sheets, in preparation, 2009.

    Google Scholar 

  7. A. Ayache, D. Wu, and Y. Xiao, Joint continuity of the local times of fractional Brownian sheets, Ann. Inst. H. Poincar´e Probab. Statist., 44 (2008), 727–748.

    Google Scholar 

  8. A. Ayache and Y. Xiao, Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets, J. Fourier Anal. Appl., 11 (2005), 407–439.

    Article  MathSciNet  MATH  Google Scholar 

  9. S.M. Berman, Local nondeterminism and local times of Gaussian processes, Indiana Univ. Math. J., 23 (1973), 69–94.

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Bierm´e, M.M. Meerschaert, and H.-P. Scheffler, Operator scaling stable random fields, Stoch. Process. Appl., 117 (2007), 312–332.

    Google Scholar 

  11. R. Cheng, A.G. Miamee, and M. Pourahmadi, On the geometry of Lp(μ) with applications to infinite variance processes, J. Aust. Math. Soc., 74 (2003), 35–42.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Cs¨org˝o, Z.-Y. Lin, and Q.-M. Shao, On moduli of continuity for local times of Gaussian processes, Stoch. Process. Appl., 58 (1995), 1–21.

    Google Scholar 

  13. M. Dozzi, Occupation density and sample path properties of N-parameter processes, In: Topics in Spatial Stochastic Processes (Martina Franca 2001), Lecture Notes in Math., 1802 (2002), 127–169, Springer, Berlin.

    Google Scholar 

  14. M. Dozzi and A.R. Soltani, Local time for stable moving average processes: H¨older conditions, Stoch. Process. Appl., 68 (1997), 195–207.

    Article  MathSciNet  MATH  Google Scholar 

  15. W. Ehm, Sample function properties of multi-parameter stable processes, Z.Wahrsch. verw. Gebiete, 56 (1981), 195–228.

    Google Scholar 

  16. D. Geman and J. Horowitz, Occupation densities, Ann. Probab., 8 (1980), 1–67.

    Article  MathSciNet  MATH  Google Scholar 

  17. J.-P. Kahane, Some Random Series of Functions, 2nd edition, Cambridge University Press, Cambridge, 1985.

    Google Scholar 

  18. D. Khoshnevisan, Multiparameter Processes: An Introduction to Random Fields, Springer, New York, 2002.

    Google Scholar 

  19. D. Khoshnevisan, D. Wu, and Y. Xiao, Sectorial local nondeterminism and the geometry of the Brownian sheet, Electron. J. Probab., 11 (2006), 817–843.

    MathSciNet  Google Scholar 

  20. D. Khoshnevisan and Y. Xiao, Images of the Brownian sheet, Trans. Amer. Math. Soc., 359 (2007), 3125–3151.

    Article  MathSciNet  MATH  Google Scholar 

  21. N. Kˆono and N.-R. Shieh, Local times and related sample path properties of certain self-similar processes, J. Math. Kyoto Univ., 33 (1993), 51–64.

    Google Scholar 

  22. G. K¨othe, Topological Vector Spaces I, Springer, New York, 1969.

    Google Scholar 

  23. J.B. Levy and M.S. Taqqu, Renewal reward processes with heavy-tailed interrenewal times and heavy-tailed rewards, Bernoulli, 6 (2000), 23–44.

    Article  MathSciNet  MATH  Google Scholar 

  24. E.H. Lieb and M. Loss, Analysis, American Mathematical Society, Rhode Island, 1997.

    Google Scholar 

  25. M. Maejima, A self-similar process with nowhere bounded sample paths, Z.Wahrsch. Verw. Gebiete, 65 (1983), 115–119.

    Article  MathSciNet  MATH  Google Scholar 

  26. D. Monrad and L.D. Pitt, Local nondeterminism and Hausdorff dimension, In: Progress in Probability and Statistics. Seminar on Stochastic Processes 1986, E. Cinlar, K.L. Chung, R.K. Getoor, Eds., Birkh¨auser, Boston, 1987, 163–189.

    Google Scholar 

  27. J.P. Nolan, Path properties of index-β stable fields, Ann. Probab., 16 (1988), 1596–1607.

    Article  MathSciNet  MATH  Google Scholar 

  28. J.P. Nolan, Local nondeterminism and local times for stable processes, Probab. Th. Rel. Fields, 82 (1989), 387–410.

    Article  MathSciNet  MATH  Google Scholar 

  29. J.P. Nolan and Y. Xiao, Geometric properties of stable random fields, preprint, 2009.

    Google Scholar 

  30. V. Pipiras and M.S. Taqqu, The limit of a renewal-reward process with heavy-tailed rewards is not a linear fractional stable motion, Bernoulli, 6 (2000), 607–614.

    Article  MathSciNet  MATH  Google Scholar 

  31. E.J.G. Pitman, On the behavior of the characteristic function of a probability distribution in the neighbourhood of the origin, J. Australian Math. Soc. Series A, 8 (1968), 422–443.

    MathSciNet  Google Scholar 

  32. L.D. Pitt, Local times for Gaussian vector fields, Indiana Univ. Math. J., 27 (1978), 309–330.

    Article  MathSciNet  MATH  Google Scholar 

  33. S.T. Rachev and S. Mittnik, Stable Paretian Models in Finance, John Wiley & Sons, Inc., 2000.

    Google Scholar 

  34. G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance, Chapman and Hall, New York, 1994.

    Google Scholar 

  35. N.-R. Shieh, Multiple points of fractional stable processes, J. Math. Kyoto Univ., 33 (1993), 731–741.

    MathSciNet  MATH  Google Scholar 

  36. N.-R. Shieh and Y. Xiao, Hausdorff and packing dimension results for self-similar random fields, submitted, 2009.

    Google Scholar 

  37. K. Takashima, Sample path properties of ergodic self-similar processes, Osaka J. Math., 26 (1989), 159–189.

    MathSciNet  MATH  Google Scholar 

  38. D.Wu and Y. Xiao, Geometric properties of the images of fractional Brownian sheets, J. Fourier Anal. Appl., 13 (2007), 1–37.

    Article  MathSciNet  MATH  Google Scholar 

  39. D. Wu and Y. Xiao, Continuity with respect to the Hurst index of the local times of anisotropic Gaussian random fields, Stoch. Process. Appl., 119 (2009), 1823–1844.

    Article  MathSciNet  MATH  Google Scholar 

  40. Y. Xiao, Dimension results for Gaussian vector fields and index-α stable fields, Ann. Probab., 23 (1995), 273–291.

    Article  MathSciNet  MATH  Google Scholar 

  41. Y. Xiao, H¨older conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields, Probab. Theory Relat. Fields, 109 (1997), 129–157.

    Article  MATH  Google Scholar 

  42. Y. Xiao, Properties of local nondeterminism of Gaussian and stable random fields and their applications, Ann. Fac. Sci. Toulouse Math., XV (2006), 157–193.

    Google Scholar 

  43. Y. Xiao, Strong local nondeterminism and sample path properties of Gaussian random fields, In: Asymptotic Theory in Probability and Statistics with Applications, Tze Leung Lai, Qiman Shao, Lianfen Qian, Eds., Higher Education Press, 2007, 136–176.

    Google Scholar 

  44. Y. Xiao, Sample path properties of anisotropic Gaussian random fields, In: A Minicourse on Stochastic Partial Differential Equations, D. Khoshnevisan and F. Rassoul-Agha, Eds., Lect. Notes in Math., 1962 (2009), 145–212, Springer, New York.

    Google Scholar 

  45. Y. Xiao, On uniform modulus of continuity of random fields, Monatsh. Math., DOI 10.1007/s00605-009-0133-z, 2009.

    Google Scholar 

  46. Y. Xiao and T. Zhang, Local times of fractional Brownian sheets, Probab. Theory Relat. Fields, 124 (2002), 204–226.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Statistics and Probability, Michigan State University, A-413 Wells Hall, East Lansing, MI, 48824, USA

    Yimin Xiao

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  1. Yimin Xiao
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Correspondence to Yimin Xiao .

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Editors and Affiliations

  1. Fac. Sciences de Base (FSB), Institut de Mathématiques (IMA), Ecole Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland

    Robert Dalang

  2. Institut Élie Cartan IECN, Nancy Université, Vandoeuvre-les-Nancy CX, France

    Marco Dozzi

  3. , Unité de Mathématiques appliquées, Ecole Nationale Supérieure des Technique, Boulevard Victor 32, Paris Cedex 15, 75739, France

    Francesco Russo

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Xiao, Y. (2011). Properties of Strong Local Nondeterminism and Local Times of Stable Random Fields. In: Dalang, R., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications VI. Progress in Probability, vol 63. Springer, Basel. https://doi.org/10.1007/978-3-0348-0021-1_18

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