Annals of Probability

Large deviations for local times and intersection local times of fractional Brownian motions and Riemann–Liouville processes

Xia Chen, Wenbo V. Li, Jan Rosiński, and Qi-Man Shao

Full-text: Open access


In this paper, we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann–Liouville processes. We also show that a fractional Brownian motion and the related Riemann–Liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times. As a consequence of our large deviation estimates, we derive laws of iterated logarithm for the corresponding local times. The key points of our methods: (1) logarithmic superadditivity of a normalized sequence of moments of exponentially randomized local time of a fractional Brownian motion; (2) logarithmic subadditivity of a normalized sequence of moments of exponentially randomized intersection local time of Riemann–Liouville processes; (3) comparison of local and intersection local times based on embedding of a part of a fractional Brownian motion into the reproducing kernel Hilbert space of the Riemann–Liouville process.

Article information

Ann. Probab., Volume 39, Number 2 (2011), 729-778.

First available in Project Euclid: 25 February 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G22: Fractional processes, including fractional Brownian motion 60J55: Local time and additive functionals 60F10: Large deviations 60G15: Gaussian processes 60G18: Self-similar processes

Local time intersection local time large deviations fractional Brownian motion Riemann–Liouville process law of iterated logarithm


Chen, Xia; Li, Wenbo V.; Rosiński, Jan; Shao, Qi-Man. Large deviations for local times and intersection local times of fractional Brownian motions and Riemann–Liouville processes. Ann. Probab. 39 (2011), no. 2, 729--778. doi:10.1214/10-AOP566.

Export citation


  • [1] Anderson, T. W. (1958). An Introduction to Multivariate Statistical Analysis. Wiley, New York.
  • [2] Asselah, A. and Castell, F. (2007). Random walk in random scenery and self-intersection local times in dimensions d≥5. Probab. Theory Related Fields 138 1–32.
  • [3] Baraka, D. and Mountford, T. (2008). A law of the iterated logarithm for fractional Brownian motions. In Séminaire de Probabilités XLI. Lecture Notes in Math. 1934 161–179. Springer, Berlin.
  • [4] Baraka, D., Mountford, T. and Xiao, Y. (2009). Hölder properties of local times for fractional Brownian motions. Metrika 69 125–152.
  • [5] Berlinet, A. and Thomas-Agnan, C. (2004). Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer, Boston, MA.
  • [6] Berman, S. M. (1969). Local times and sample function properties of stationary Gaussian processes. Trans. Amer. Math. Soc. 137 277–299.
  • [7] Berman, S. M. (1973/74). Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23 69–94.
  • [8] Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.
  • [9] Chen, X. (2004). Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks. Ann. Probab. 32 3248–3300.
  • [10] Chen, X. (2010). Random Walk Intersections: Large Deviations and Related Topics. Mathematical Surveys and Monographs 157. Amer. Math. Soc., Providence, RI.
  • [11] Chen, X. and Li, W. V. (2003). Quadratic functionals and small ball probabilities for the m-fold integrated Brownian motion. Ann. Probab. 31 1052–1077.
  • [12] Chen, X. and Li, W. V. (2004). Large and moderate deviations for intersection local times. Probab. Theory Related Fields 128 213–254.
  • [13] Donsker, M. D. and Varadhan, S. R. S. (1981). The polaron problem and large deviations. Phys. Rep. 77 235–237.
  • [14] Fernández, R., Fröhlich, J. and Sokal, A. D. (1992). Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory. Springer, Berlin.
  • [15] Fleischmann, K., Mörters, P. and Wachtel, V. (2008). Moderate deviations for a random walk in random scenery. Stochastic Process. Appl. 118 1768–1802.
  • [16] Gantert, N., König, W. and Shi, Z. (2007). Annealed deviations of random walk in random scenery. Ann. Inst. H. Poincaré Probab. Statist. 43 47–76.
  • [17] Geman, D., Horowitz, J. and Rosen, J. (1984). A local time analysis of intersections of Brownian paths in the plane. Ann. Probab. 12 86–107.
  • [18] Gradshteyn, I. S. and Ryzhik, I. M. (2000). Table of Integrals, Series, and Products, 6th ed. Academic Press, San Diego, CA.
  • [19] Hamana, Y. and Kesten, H. (2001). A large-deviation result for the range of random walk and for the Wiener sausage. Probab. Theory Related Fields 120 183–208.
  • [20] van der Hofstad, R., König, W. and Mörters, P. (2006). The universality classes in the parabolic Anderson model. Comm. Math. Phys. 267 307–353.
  • [21] den Hollander, F. (2009). Random Polymers. Lecture Notes in Math. 1974. Springer, Berlin.
  • [22] Hu, Y. and Nualart, D. (2005). Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab. 33 948–983.
  • [23] Hu, Y., Nualart, D. and Song, J. (2008). Integral representation of renormalized self-intersection local times. J. Funct. Anal. 255 2507–2532.
  • [24] König, W. and Mörters, P. (2002). Brownian intersection local times: Upper tail asymptotics and thick points. Ann. Probab. 30 1605–1656.
  • [25] Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Boston, MA.
  • [26] Le Gall, J. F. (1986). Propriétés d’intersection des marches aléatoires. I. Convergence vers le temps local d’intersection. Comm. Math. Phys. 104 471–507.
  • [27] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces: Isoperimetry and Processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 23. Springer, Berlin.
  • [28] Li, W. V. and Linde, W. (1998). Existence of small ball constants for fractional Brownian motions. C. R. Acad. Sci. Paris Sér. I Math. 326 1329–1334.
  • [29] Li, W. V. and Linde, W. (1999). Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Probab. 27 1556–1578.
  • [30] Li, W. V. and Shao, Q. M. (2001). Gaussian processes: Inequalities, small ball probabilities and applications. In Stochastic Processes: Theory and Methods. Handbook of Statist. 19 533–597. North-Holland, Amsterdam.
  • [31] Madras, N. and Slade, G. (1993). The Self-Avoiding Walk. Birkhäuser, Boston, MA.
  • [32] Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422–437.
  • [33] Marcus, M. B. and Rosen, J. (1997). Laws of the iterated logarithm for intersections of random walks on Z4. Ann. Inst. H. Poincaré Probab. Statist. 33 37–63.
  • [34] Mishura, Y. S. (2008). Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Math. 1929. Springer, Berlin.
  • [35] Nualart, D. and Ortiz-Latorre, S. (2007). Intersection local time for two independent fractional Brownian motions. J. Theoret. Probab. 20 759–767.
  • [36] Pipiras, V. and Taqqu, M. S. (2002). Deconvolution of fractional Brownian motion. J. Time Ser. Anal. 23 487–501.
  • [37] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [38] Rosen, J. (1987). The intersection local time of fractional Brownian motion in the plane. J. Multivariate Anal. 23 37–46.
  • [39] Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Gordon & Breach, Yverdon.
  • [40] van der Vaart, A. W. and van Zanten, J. H. (2008). Reproducing kernel Hilbert spaces of Gaussian priors. In Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh. Inst. Math. Stat. Collect. 3 200–222. IMS, Beachwood, OH.
  • [41] Wu, D. and Xiao, Y. (2009). Regularity of intersection local times of fractional Brownian motions. J. Theor. Probab. 23 972–1001.
  • [42] Xiao, Y. (2008). Strong local nondeterminism and sample path properties of Gaussian random fields. In Asymptotic Theory in Probability and Statistics with Applications. Adv. Lectures Math. (ALM) 2 136–176. Internetional Press, Somerville, MA.