Annals of Applied Probability

Efficient Monte Carlo for high excursions of Gaussian random fields

Robert J. Adler, Jose H. Blanchet, and Jingchen Liu

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Our focus is on the design and analysis of efficient Monte Carlo methods for computing tail probabilities for the suprema of Gaussian random fields, along with conditional expectations of functionals of the fields given the existence of excursions above high levels, b. Naïve Monte Carlo takes an exponential, in b, computational cost to estimate these probabilities and conditional expectations for a prescribed relative accuracy. In contrast, our Monte Carlo procedures achieve, at worst, polynomial complexity in b, assuming only that the mean and covariance functions are Hölder continuous. We also explain how to fine tune the construction of our procedures in the presence of additional regularity, such as homogeneity and smoothness, in order to further improve the efficiency.

Article information

Ann. Appl. Probab., Volume 22, Number 3 (2012), 1167-1214.

First available in Project Euclid: 18 May 2012

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Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 65C05: Monte Carlo methods
Secondary: 60G60: Random fields 62G32: Statistics of extreme values; tail inference

Gaussian random fields high-level excursions Monte Carlo tail distributions efficiency


Adler, Robert J.; Blanchet, Jose H.; Liu, Jingchen. Efficient Monte Carlo for high excursions of Gaussian random fields. Ann. Appl. Probab. 22 (2012), no. 3, 1167--1214. doi:10.1214/11-AAP792.

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  • [1] Adler, R. J. (1981). The Geometry of Random Fields. Wiley, Chichester.
  • [2] Adler, R. J., Bartz, K. and Kou, S. (2010). Estimating curvatures in the Gaussian kinematic formula. Unpublished manuscript, Harvard Univ.
  • [3] Adler, R. J., Blanchet, J. and Liu, J. (2008). Efficient simulation for tail probabilities of Gaussian random fields. In WSC’08: Proceedings of the 40th Conference on Winter Simulation 328–336. Winter Simulation Conference, Austin, TX.
  • [4] Adler, R. J., Müller, P. and Rozovskiĭ, B., eds. (1996). Stochastic Modelling in Physical Oceanography. Progress in Probability 39. Birkhäuser, Boston, MA.
  • [5] Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
  • [6] Adler, R. J., Taylor, J. E. and Worsley, K. (2011). Applications of random fields and geometry: Foundations and case studies. Available at
  • [7] Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Stochastic Modelling and Applied Probability 57. Springer, New York.
  • [8] Azaïs, J.-M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. Wiley, Hoboken, NJ.
  • [9] Bardeen, J. M., Bond, J. R., Kaiser, N. and Szalay, A. S. (1986). The statistics of peaks of Gaussian random fields. The Astrophysical Journal 304 15–61.
  • [10] Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA.
  • [11] Blanchet, J. H. (2009). Efficient importance sampling for binary contingency tables. Ann. Appl. Probab. 19 949–982.
  • [12] Borell, C. (1975). The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30 207–216.
  • [13] Bucklew, J. A. (2004). Introduction to Rare Event Simulation. Springer, New York.
  • [14] Cirel’son, B. S., Ibragimov, I. A. and Sudakov, V. N. (1976). Norms of Gaussian sample functions. In Proceedings of the Third Japan–USSR Symposium on Probability Theory (Tashkent, 1975). Lecture Notes in Math. 550 20–41. Springer, Berlin.
  • [15] Dennis, M. R. (2007). Nodal densities of planar Gaussian random waves. Eur. Phys. J. 145 191–210.
  • [16] Dudley, R. M. (1973). Sample functions of the Gaussian process. Ann. Probab. 1 66–103.
  • [17] Friston, K. J., Worsley, K. J., Frackowiak, R. S. J., Mazziotta, J. C. and Evans, A. C. (1994). Assessing the significance of focal activations using their spatial extent. Human Brain Mapping 1 214–220.
  • [18] Kallenberg, O. (1986). Random Measures, 4th ed. Akademie Verlag, Berlin.
  • [19] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [20] Landau, H. J. and Shepp, L. A. (1970). On the supremum of a Gaussian process. Sankhyā Ser. A 32 369–378.
  • [21] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • [22] Mitzenmacher, M. and Upfal, E. (2005). Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge Univ. Press, Cambridge.
  • [23] Niemiro, W. and Pokarowski, P. (2009). Fixed precision MCMC estimation by median of products of averages. J. Appl. Probab. 46 309–329.
  • [24] Piterbarg, V. (1995). Asymptotic Methods in the Theory of Gaussian Processes. Amer. Math. Soc., Providence, RI.
  • [25] Shandarin, S. F. (2002). Testing non-Gaussianity in cosmic microwave background maps by morphological statistics. Mon. Not. R. Astr. Soc. 331 865–874.
  • [26] Shandarin, S. F., Feldman, H. A., Xu, Y. and Tegmark, M. (2002). Morphological measures of non-Gaussianity in cosmic microwave background maps. Astrophys. J. Suppl. 141 1–11.
  • [27] Taylor, J., Takemura, A. and Adler, R. J. (2005). Validity of the expected Euler characteristic heuristic. Ann. Probab. 33 1362–1396.
  • [28] Taylor, J. E. and Worsley, K. J. (2007). Detecting sparse signals in random fields, with an application to brain mapping. J. Amer. Statist. Assoc. 102 913–928.
  • [29] Traub, J. F., Wasilkowski, G. W. and Woźniakowski, H. (1988). Information-Based Complexity. Academic Press, Boston, MA.
  • [30] Tsirel’son, V. S. (1975). The density of the maximum of a Gaussian process. Theory Probab. Appl. 20 847–856.
  • [31] Woźniakowski, H. (1996). Computational complexity of continuous problems. Technical report, Columbia Univ.