• Bernoulli
  • Volume 16, Number 4 (2010), 926-952.

Hausdorff and packing dimensions of the images of random fields

Narn-Rueih Shieh and Yimin Xiao

Full-text: Open access


Let $X = \{X(t), t ∈ ℝ^N\}$ be a random field with values in $ℝ^d$. For any finite Borel measure $μ$ and analytic set $E ⊂ ℝ^N$, the Hausdorff and packing dimensions of the image measure $μ_X$ and image set $X(E)$ are determined under certain mild conditions. These results are applicable to Gaussian random fields, self-similar stable random fields with stationary increments, real harmonizable fractional Lévy fields and the Rosenblatt process.

Article information

Bernoulli, Volume 16, Number 4 (2010), 926-952.

First available in Project Euclid: 18 November 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Hausdorff dimension images packing dimension packing dimension profiles real harmonizable fractional Lévy motion Rosenblatt process self-similar stable random fields


Shieh, Narn-Rueih; Xiao, Yimin. Hausdorff and packing dimensions of the images of random fields. Bernoulli 16 (2010), no. 4, 926--952. doi:10.3150/09-BEJ244.

Export citation


  • Adler, R.J. (1981). The Geometry of Random Fields. New York: Wiley.
  • Albin, J.M.P. (1998a). A note on the Rosenblatt distributions. Statist. Probab. Lett. 40 83–91.
  • Albin, J.M.P. (1998b). On extremal theory for self-similar processes. Ann. Probab. 26 743–793.
  • Benassi, A., Cohen, S. and Istas, J. (2002). Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8 97–115.
  • Benassi, A., Cohen, S. and Istas, J. (2003). Local self-similarity and the Hausdorff dimension. C. R. Math. Acad. Sci. Paris 336 267–272.
  • Benassi, A., Cohen, S. and Istas, J. (2004). On roughness indices for fractional fields. Bernoulli 10 357–373.
  • Biermé, H. and Lacaux, C. (2009). Hölder regularity for operator scaling stable random fields. Stochastic Process. Appl. 119 2222–2248.
  • Csáki, E. and Csörgő, M. (1992). Inequalities for increments of stochastic processes and moduli of continuity. Ann. Probab. 20 1031–1052.
  • Davydov, Yu.A. (1990). On distributions of multiple Wiener–Itô integrals. Theory Probab. Appl. 35 27–37.
  • Dobrushin, R.L. and Major, P. (1979). Non-central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 27–52.
  • Ehm, W. (1981). Sample function properties of multi-parameter stable processes. Z. Wahrsch. Verw. Gebiete 56 195–228.
  • Falconer, K.J. (1990). Fractal Geometry – Mathematical Foundations and Applications. New York: Wiley.
  • Falconer, K.J. and Howroyd, J.D. (1997). Packing dimensions for projections and dimension profiles. Math. Proc. Cambridge Philos. Soc. 121 269–286.
  • Howroyd, J.D. (2001). Box and packing dimensions of projections and dimension profiles. Math. Proc. Cambridge Philos. Soc. 130 135–160.
  • Hu, X. and Taylor, S.J. (1994). Fractal properties of products and projections of measures in ℝd. Math. Proc. Cambridge Philos. Soc. 115 527–544.
  • Kahane, J.-P. (1985). Some Random Series of Functions, 2nd ed. Cambridge: Cambridge Univ. Press.
  • Khoshnevisan, D., Schilling, R. and Xiao, Y. (2009). Packing dimension profiles and Lévy processes. Preprint.
  • Khoshnevisan, D. and Xiao, Y. (2008a). Packing dimension of the range of a Lévy process. Proc. Amer. Math. Soc. 136 2597–2607.
  • Khoshnevisan, D. and Xiao, Y. (2008b). Packing dimension profiles and fractional Brownian motion. Math. Proc. Cambridge Philos. Soc. 145 205–213.
  • Kokoszka, P.S. and Taqqu, M.S. (1994). New classes of self-similar symmetric stable random fields. J. Theoret. Probab. 7 527–549.
  • Kôno, N. (1986). Hausdorff dimension of sample paths for self-similar processes. In Dependence in Probability and Statistics (Oberwolfach, 1985). Progr. Probab. Statist. 11 109–117. Boston: Birkhäuser Boston.
  • Kôno, N. and Maejima, M. (1991). Hölder continuity of sample paths of some self-similar stable processes. Tokyo J. Math. 14 93–100.
  • Levy, J.B. and Taqqu, M.S. (2000). Renewal reward processes with heavy-tailed interrenewal times and heavy-tailed rewards. Bernoulli 6 23–44.
  • Lin, H. and Xiao, Y. (1994). Dimension properties of the sample paths of self-similar processes. Acta Math. Sinica 10 289–300.
  • Lubin, A. (1974). Extensions of measures and the Von Nermann selection theorem. Proc. Amer. Math. Soc. 43 118–122.
  • Maejima, M. (1983). A self-similar process with nowhere bounded sample paths. Z. Wahrsch. Verw. Gebiete 65 115–119.
  • Major, P. (1981). Multiple Wiener–Itô Integrals. Lecture Notes in Math. 849. Berlin: Sringer.
  • Mattila, P. (1995). Geometry of Sets and Measures in Euclidean Spaces. Cambridge: Cambridge Univ. Press.
  • Mori, T. and Oodaira, H. (1986). The law of the iterated logarithm for self-similar processes represented by multiple Wiener integrals. Probab. Theory Related Fields 71 367–391.
  • Móricz, F., Serfling, R.J. and Stout, W.F. (1982). Moment and probability bounds with quasi superadditive structure for the maximum partial sum. Ann. Probab. 10 1032–1040.
  • Nolan, J. (1989). Continuity of symmetric stable processes. J. Multivariate Anal. 29 84–93.
  • Pipiras, V. (2004). Wavelet-type expansion of the Rosenblatt process. J. Fourier Anal. Appl. 10 599–634.
  • Pipiras, V. and Taqqu, M.S. (2000). The limit of a renewal-reward process with heavy-tailed rewards is not a linear fractional stable motion. Bernoulli 6 607–614.
  • Rosinski, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Probab. 21 996–1104.
  • Samorodnitsky, G. and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. New York: Chapman & Hall.
  • Schilling, R.L. (1998). Feller processes generated by pseudo-differential operators: On the Hausdorff dimension of their sample paths. J. Theoret. Probab. 11 303–330.
  • Schilling, R.L. and Xiao, Y. (2009). Packing dimension of the images of Markov processes. Preprint.
  • Shieh, N.-R. (1996). Sample functions of Lévy–Chentsov random fields. In Probability Theory and Mathematical Statistics. Proceedings of the 7th Japan–Russia Symposium (S. Watanabe et al., eds.) 450–459. River Edge, NJ: World Scientific.
  • Takashima, K. (1989). Sample path properties of ergodic self-similar processes. Osaka Math. J. 26 159–189.
  • Talagrand, M. and Xiao, Y. (1996). Fractional Brownian motion and packing dimension. J. Theoret. Probab. 9 579–593.
  • Taqqu, M.S. (1975). Weak convergence to farctional Brownian motion and to Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287–302.
  • Taqqu, M.S. (1979). Convergence to integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 53–83.
  • Taylor, S.J. (1986). The measure theory of random fractals. Math. Proc. Cambridge Philos. Soc. 100 383–406.
  • Taylor, S.J. and Tricot, C. (1985). Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288 679–699.
  • Tricot, C. (1982). Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc. 91 57–74.
  • Tudor, C.A. (2008). Analysis of the Rosenblatt process. ESAIM Probab. Statist. 12 230–257.
  • Xiao, Y. (1997). Packing dimension of the image of fractional Brownian motion. Statist. Probab. Lett. 33 379–387.
  • Xiao, Y. (2004). Random fractals and Markov processes. In Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot (M.L. Lapidus and M. van Frankenhuijsen, eds.) 261–338. Providence, RI: Amer. Math. Soc.
  • Xiao, Y. (2007). Strong local nondeterminism and the sample path properties of Gaussian random fields. In Asymptotic Theory in Probability and Statistics with Applications (T.L. Lai, Q. Shao and L. Qian, eds.) 136–176. Beijing: Higher Education Press.
  • Xiao, Y. (2009). A packing dimension theorem for Gaussian random fields. Statist. Probab. Lett. 79 88–97.
  • Xiao, Y. (2010). On uniform modulus of continuity of random fields. Monatsh. Math. 159 163–184.