The Annals of Probability

Lower tail probabilities for Gaussian processes

Wenbo V. Li and Qi-Man Shao

Full-text: Open access


Let $X=(X_t)_{t \in S}$ be a real-valued Gaussian random process indexed by S with mean zero. General upper and lower estimates are given for the lower tail probability $\mathbb{P}(\sup_{t \in S} (X_t-X_{t_0}) \leq x )$ as $x \to 0$, with $t_0\in S$ fixed. In particular, sharp rates are given for fractional Brownian sheet. Furthermore, connections between lower tail probabilities for Gaussian processes with stationary increments and level crossing probabilities for stationary Gaussian processes are studied. Our methods also provide useful information on a random pursuit problem for fractional Brownian particles.

Article information

Ann. Probab., Volume 32, Number 1A (2004), 216-242.

First available in Project Euclid: 4 March 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes
Secondary: 60G17: Sample path properties 60G60: Random fields

Lower tail probability Gaussian processes


Li, Wenbo V.; Shao, Qi-Man. Lower tail probabilities for Gaussian processes. Ann. Probab. 32 (2004), no. 1A, 216--242. doi:10.1214/aop/1078415834.

Export citation


  • Bass, R., Eisenbaum, N. and Shi, Z. (2000). The most visited sites of symmetric stable processes. Probab. Theory Related Fields 116 391--404.
  • Bellman, R. (1970). Introduction to Matrix Analysis. McGraw-Hill, New York. [Reprinted (1997) by SIAM.]
  • Bramson, M. and Griffeath, D. (1991). Capture problems for coupled random walks. In Random Walks, Brownian Motion and Interacting Particle Systems (R. Durrett and H. Kesten, eds.) 153--188. Birkhäuser, Boston.
  • Csáki, E., Khoshnevisan, D. and Shi, Z. (2000). Boundary crossings and the distribution function of the maximum of Brownian sheet. Stochastic Process. Appl. 90 1--18.
  • Dembo, A., Poonen, B., Shao, Q.-M. and Zeitouni, O. (2002). Random polynomials having few or no real zeros. J. Amer. Math. Soc. 15 857--892.
  • Dudley, R. M. (1967). The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1 290--330.
  • Fernique, X. (1964). Continuité des processus Gaussiens. C. R. Acad. Sci. Paris 258 6058--6060.
  • Kesten, H. (1991). An absorption problem for several Brownian motions. In Seminar on Stochastic Processes (E. Çinlar, K. L. Chung and M. J. Sharpe, eds.) 59--72. Birkhäuser, Boston.
  • Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • Ledoux, M. (1996). Isoperimetry and Gaussian analysis. Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1648 165--294. Springer, New York.
  • Ledoux, M. and Talagrand, M. (1991). Probability on Banach Spaces. Springer, New York.
  • Li, W. V. and Shao, Q.-M. (2001a). Gaussian processes: Inequalities, small ball probabilities and applications. In Stochastic Processes: Theory and Methods (C. R. Rao and D. Shanbhag, eds.) 533--598. North-Holland, Amsterdam.
  • Li, W. V. and Shao, Q.-M. (2001b). Capture time of Brownian pursuits. Probab. Theory Related Fields 121 30--48.
  • Li, W. V. and Shao, Q.-M. (2002). A normal comparison inequality and its applications. Probab. Theory Related Fields 122 494--508.
  • Marcus, M. (2000). Probability estimates for lower levels of certain Gaussian processes with stationary increments. In High Dimensional Probability II (E. Gine, D. Mason and J. Wellner, eds.) 173--179. Birkhäuser, Boston.
  • Molchan, M. (1999). Maximum of a fractional Brownian motion: Probabilities of small values. Comm. Math. Phys. 205 97--111.
  • Molchan, M. (2000). On the maximum of fractional Brownian motion. Theory Probab. Appl. 44 97--102.
  • Newell, G. and Rosenblatt, M. (1962). Zero crossing probabilities for Gaussian stationary processes. Ann. Math. Stat. 33 1306--1313.
  • Pickands, J. (1969). Asymptotic properties of maximum in a stationary Gaussian processes. Trans. Amer. Math. Soc. 145 75--86.
  • Price, G. (1951). Bounds for determinants with dominant principal diagonal. Proc. Amer. Math. Soc. 2 497--502.
  • Shao, Q.-M. (1999). A Gaussian correlation inequality and its applications to the existence of small ball constant. Preprint.
  • Slepian, D. (1961). First passage time for a particular Gaussian process. Ann. Math. Statist. 32 610--612.
  • Slepian, D. (1962). The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41 463--501.
  • Strakhov, N. and Kurz, L. (1968). An upper bound on the zero-crossing distribution. Bell System Tech. J. 47 529--547.