The Annals of Applied Probability

On excursion sets, tube formulas and maxima of random fields

Robert J. Adler

Full-text: Open access


This is a rambling review of what, with a few notable and significant exceptions, has been a rather dormant area for over a decade. It concentrates on the septuagenarian problem of finding good approximations for the excursion probability $P\{sup_{t \in T}X_t \geq \lambda\}$, where $\lambda$ is large, $X$ is a Gaussian, or “Gaussian-like,” process over a region $T \subset \Re^N$, and, generally, $N > 1$. A quarter of a century ago, there was a flurry of papers out of various schools linking this problem to the geometrical properties of random field sample paths. My own papers made the link via Euler characteristics of the excursion sets $\{t \in T: X_t \geq \lambda\}$. A decade ago, Aldous popularized the Poisson clumping heuristic for computing excursion probabilities in a wide variety of scenarios, including the Gaussian. Over the past few years, Keith Worsley has been the driving force behind the computation of many new Euler characteristic functionals, primarily driven by applications in medical imaging. There has also been a parallel development of techniques in the astrophysical literature. Meanwhile, somewhat closer to home, Hotelling’s 1939 “tube formulas” have seen a renaissance as sophisticated statistical hypothesis testing problems led to their reapplication toward computing excursion probabilities, and Sun and others have shown how to apply them in a purely Gaussian setting. The aim of the present paper is to look again at many of these results and tie them together in new ways to obtain a few new results and, hopefully, considerable new insight. The “Punchline of this paper,”which relies heavily on a recent result of Piterbarg, is given in Section 6.6: “In computing excursion probabilities for smooth enough Gaussian random fields over reasonable enough regions, the expected Euler characteristic of the corresponding excursion sets gives an approximation, for large levels, that is accurate to as many terms as there are in its expansion.”

Article information

Ann. Appl. Probab. Volume 10, Number 1 (2000), 1-74.

First available: 25 April 2002

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60G60: Random fields 60G70: Extreme value theory; extremal processes
Secondary: 60G17: Sample path properties 62M40: Random fields; image analysis 62H10: Distribution of statistics

Random fields excursion sets tube formulas extremal distributions maxima Euler characteristic


Adler, Robert J. On excursion sets, tube formulas and maxima of random fields. The Annals of Applied Probability 10 (2000), no. 1, 1--74. doi:10.1214/aoap/1019737664.

Export citation


  • [1] Adler, R. J. (1981). The Geometry of Random Fields. Wiley, London.
  • [2] Adler, R. J. (1990). An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. IMS, Hayward, CA.
  • [3] Adler, R. J. (2001). Random Fields and Their Geometry. Birkh¨auser, Boston.
  • [4] Adler, R. J. and Firman, D. (1981). A non-Gaussian model for random surfaces. Philos. Trans. Roy. Soc. London Ser. A 303 433-462.
  • [5] Adler, R. J. and Samorodnitsky, G. (1997). Level crossings of absolutely continuous stationary symmetric -stable processes. Ann. Appl. Probab. 6 460-493.
  • [6] Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.
  • [7] Aza¨is, J.-M. and Wschebor, M. (1999). The distribution of the maximum of a stochastic process by the Rice method. Preprint.
  • [8] Bennett et al. (1994). Morphology of the interstellar cooling lines detected by COBE. Astrophys. J. 434 587-598.
  • [9] Berman, S. M. (1964). Limit theorems for the maximum term in stationary sequences. Ann. Math. Statist. 35 502-516.
  • [10] Berman, S. M. (1971). Maxima and high level excursions of stationary Gaussian processes. Trans. Amer. Math. Soc. 160 65-85.
  • [11] Berman, S. M. (1971). Asymptotic independence of the numbers of high and low level crossings of stationary Gaussian processes. Ann. Math. Statist. 42 927-945.
  • [12] Berman, S. M. (1992). Sojourns and Extremes of Stationary Processes. Wadsworth and Brooks/Cole, Pacific Grove, CA.
  • [13] Bickel, P. and Rosenblatt, M. (1973). Two-dimensional random fields. In Multivariate Analysis III (P. R. Krishnaiah, ed.) 3-15. Academic, New York.
  • [14] Borell, C. (1975). The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 205-216.
  • [15] Breitung, K. (1996). Higher order approximations for maxima of random fields. In IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics (A. Naess and S. Krenk, eds.) 79-88. Kluwer, Dordrecht.
  • [16] Bulinskaya, E. V. (1961). On the mean number of crossings of a level by a stationary Gaussian process. Theory Probab. Appl. 6 435-438.
  • [17] Cao, J. (2000). The size of the connected components of excursion sets of 2, t and F fields. Adv. in Appl. Probab. 31 579-595.
  • [18] Cao, J. and Worsley, K. J. (1999). The detection of local shape changes via the geometry of Hotelling's T2 fields. Ann. Statist. 27 925-942.
  • [19] Cao, J. and Worsley, K. J. (1999). The geometry of correlation fields with an application to functional connectivity of the brain. Ann. Appl. Prob. 9 1021-1057.
  • [20] Cirel´son, B. S. (1975). Density of the distribution of the maximum of a Gaussian process (in Russian). Theory Probab. Appl. 20 847-856.
  • [21] 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. Lecture Notes in Math. 550 20-41. Springer, New York.
  • [22] Cram´er, H. (1965). A limit theorem for the maximum values of certain stochastic processes. Theory Probab. Appl. 10 137.
  • [23] Cram´er, H. (1996). On the intersections between the trajectories of a normal stationary stochastic process and a high level. Ark. Mat. 6 337-349.
  • [24] Cram´er, H. and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes. Wiley, New York.
  • [25] Cuzik, J. (1976). A central limit theorem for the number of zeros of a stationary Gaussian process. Ann. Probab. 4 547-556.
  • [26] Delmas, C. (1998). An asymptotic expansion for the distribution of the maximum of a class of Gaussian fields. C. R. Acad. Sci. Paris S´er. I Math. 327 393-397.
  • [27] Diebolt, J. (1981). Sur la loi du maximum de certains processus gaussiens sur le tore. Ann. Inst. H. Poincar´e Probab. Statist. 17 165-179.
  • [28] Diebolt, J. and Posse, C. (1995). A nonasymptotic approach to the density of the maximum of smooth Gaussian processes. C. R. Acad. Sci. Paris S´er. I Math. 321 933-938.
  • [29] Diebolt, J. and Posse, C. (1996). On the density of the maximum of smooth Gaussian processes. Ann. Probab. 24 1104-1129.
  • [30] Dudley, R. M. (1967). The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1 290-330.
  • [31] Dudley, R. M. (1973). Sample functions of the Gaussian process. Ann. Probab. 1 66-103.
  • [32] Eilenberg, S. and Steenrod, N. (1952). Foundations of Algebraic Topology. Princeton Univ. Press.
  • [33] Fatalov, V. R. (1992). Exact asymptotics of large deviations of Gaussian measures in a Hilbert space (in Russian). Izv. Nats. Akad. Nauk Armenii Mat. 27 43-61.
  • [34] Fatalov, V. R. (1993). Asymptotics of the probabilities of large deviations of Gaussian fields: applications (in Russian). Izv. Nats. Akad. Nauk Armenii Mat. 28 25-51.
  • [35] Gnedenko, B.V. (1943). Sur la distribution limite du terme maximum d'une s´erie al´eatoire. Ann. Math. 44 423-453.
  • [36] Gray, A. (1990). Tubes. Addison-Wesley, Redwood City, CA.
  • [37] Hadwiger, H. (1957). Vorles ¨ungen ¨Uber Inhalt, Oberfl¨ache und Isoperimetrie. Springer, Berlin.
  • [38] Hadwiger, H. (1971). Normale K¨orper im euklidischen Raum und ihre topologischen und metrischen Eigenschaften. Math. Z. 71 124-140.
  • [39] Hall, P. (1988). Introduction to the Theory of Coverage Processes. Wiley, New York.
  • [40] Hogan, M. L. and Siegmund, D. (1986). Large deviations for the maxima of some random fields. Adv. in Appl. Math. 7 2-22.
  • [41] Hotelling, H. (1939). Tubes and spheres in n-spaces and a class of statistical problems. Amer. J. Math. 61 440-460.
  • [42] It o, K. (1964). The expected number of zeros of continuous stationary Gaussian processes. J. Math. Kyoto Univ. 3 206-216.
  • [43] James, B., James, K. L. and Siegmund, D. (1988). Conditional boundary crossing probabilities, with applications to change-point problems. Ann. Probab. 16 825-839.
  • [44] Jennen, C. (1985). Second-order approximations to the density, mean and variance of Brownian first-exit times. Ann. Probab. 13 126-144.
  • [45] Johansen, S. and Johnstone, I. M. (1990). Hotelling's theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis. Ann. Statist. 18 652- 684.
  • [46] Johnstone, I. and Siegmund, D. (1989). On Hotelling's formula for the volume of tubes and Naiman's inequality. Ann. Statist. 17 184-194.
  • [47] Kac, M. (1943). On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc. 43 314-320.
  • [48] Kac, M. and Slepian, D. (1959). Large excursions of Gaussian processes. Ann. Math. Statist. 30 1215-1228.
  • [49] Knowles, M. and Siegmund, D. (1989). On Hotelling's approach to testing for a nonlinear parameter in a regression. Internat. Statist. Rev. 57 205-220.
  • [50] Kratz, M. and Rootz´en, H. (1997). On the rate of convergence for extremes of mean square differentiable stationary normal processes. J. Appl. Probab. 34 908-923.
  • [51] Kreyszig, E. (1968). Introduction to Differential Geometry and Reimannian Geometry. Univ. Toronto Press. Toronto.
  • [52] Landau, H. and Shepp, L. A. (1970). On the supremum of a Gaussian process. Sankhy¯a Ser. A 32 369-378.
  • [53] Leadbetter, M. R., Lindgren, G. and Rootz´en, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • [54] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Isoperimetry and Processes. Springer, Berlin.
  • [55] Lerche, H. R. and Siegmund, D. (1989). Approximate exit probabilities for a Brownian bridge on a short time interval, and applications. Adv. in Appl. Probab. 21 1-19.
  • [56] Lifshits, M. A. (1994). Tail probabilities of Gaussian extrema and Laplace transform. Ann. Inst. H. Poincar´e Probab. Statist. 30 163-179.
  • [57] Lindgren, G. (1972). Local maxima of Gaussian fields. Ark. Mat. 10 195-218.
  • [58] Malevich, T. L. (1969). Asymptotic normality of the number of crossings of level zero by a Gaussian process. Theory Probab. Appl. 14 287-295.
  • [59] Marcus, M. B. (1987). -Radial Processes and Random Fourier Series. Amer. Math. Soc., Providence, RI.
  • [60] Marcus, M. B. (1989). Some bounds for the expected number of level crossings of symmetric harmonizable p-stable processes. Stochastic Process. Appl. 33 217-231.
  • [61] Marcus, M. B. and Pisier, G. (1981). Random Fourier Series with Applications to Harmonic Analysis. Princeton Univ. Press.
  • [62] Marcus, M. B. and Shen, K. (1997). Bounds for the expected number of level crossings of certain harmonizable infinitely divisible processes. Preprint.
  • [63] Marcus, M. B. and Shepp, L. A. (1971). Sample behavior of Gaussian processes. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2 423-442. Univ. California Press, Berkeley.
  • [64] Matheron, G. (1975). Random Sets and Integral Geometry. Wiley, New York.
  • [65] McCormick, W. P. (1997). A geometric approach to obtaining the distribution of the maximum for a class of random fields. Preprint.
  • [66] Mikhaleva, T. L. and Piterbarg, V. I. (1996). On the distribution of the maximum of a Gaussian field with constant variance on a smooth manifold. Theory Probab. Appl. 41 367-379.
  • [67] Millman, R. S. and Parker, G. D. (1977). Elements of Differential Geometry. Prentice-Hall, Englewood Cliffs, NJ.
  • [68] Morse, M. and Cairns, S. (1969). Critical Point Theory in Global Analysis and Differential Topology. Academic, New York.
  • [69] Naiman, D. Q. (1990). Volumes of tubular neighborhoods of spherical polyhedra and statistical inference. Ann. Statist. 18 685-716.
  • [70] Naiman, D. Q. and Wynn, H. P. (1992). Inclusion-exclusion-Bonferroni identities and inequalities for discrete tube-like problems via Euler characteristics. Ann. Statist. 20 43-76.
  • [71] Pickands, J., III (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145 51-74.
  • [72] Pickands, J., III (1969). Asymptotic properties of the maximum in a stationary Gaussian process. Trans. Amer. Math. Soc. 145 75-86.
  • [73] Pisier, G. (1986). Probabilistic methods in the geometry of Banach space. Lecture Notes in Math. 1206 167-241. Springer, New York.
  • [74] Piterbarg, V. I. (1981). Comparison of distribution functions for maxima of Gaussian processes. Theory Probab. Appl. 26 687-705.
  • [75] Piterbarg, V. I. (1982). Large deviations of random processes close to Gaussian ones. Theory Probab. Appl. 27 504-524.
  • [76] Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. Amer. Math. Soc., Providence, RI.
  • [77] Piterbarg, V. I. (1997). Rice's method for large excursions of Gaussian random fields. Technical Report 478, Center for Stochastic Processes, Univ. North Carolina. [Translation of Russian version in Fund. Appl. Math. 2 (1996) 187-204.]
  • [78] Piterbarg, V. I. and Stamatovich, S. (1998). On the maximum of Gaussian non-centered fields indexed on smooth manifolds. Preprint.
  • [79] Piterbarg, V. I. and Weber, M. (1997). Tail distribution results for Gaussian supremastandard methods. Technical Report 491, Center for Stochastic Processes, Univ. North Carolina.
  • [80] Powell, C. S. (1992). The golden age of cosmology. Scientific American July 17-22.
  • [81] Qualls, C. and Watanabe, H. (1973). Asymptotic properties of Gaussian random fields. Trans. Amer. Math. Soc. 177 155-171.
  • [82] Rabinowitz, D. and Siegmund, D. (1997). The approximate distribution of the maximum of a smoothed Poisson random field. Statist. Sinica 7 167-180.
  • [83] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
  • [84] Rice, S. O. (1945). Mathematical analysis of random noise. Bell. System Tech. J. 24 46-156.
  • [Reprinted in (1954). Selected Papers on Noise and Stochastic Processes (N. Wax, ed.) Dover, New York.]
  • [85] Riesz, R. and Sz-Nagy, B. (1955). Functional Analysis. Ungar, New York.
  • [86] Samorodnitsky, G. (1987). Bounds on the supremum distribution of Gaussian processespolynomial entropy case. Preprint.
  • [87] Samorodnitsky, G. (1987). Bounds on the supremum distribution of Gaussian processesexponential entropy case. Preprint.
  • [88] Samorodnitsky, G. (1991). Probability tails of Gaussian suprema. Stochastic. Process. Appl. 38 55-84.
  • [89] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.
  • [90] Santal ´o, L. A. (1976). Integral Geometry and Geometric Probability. Encyclopedia of Mathematics and Its Applications 1. Addison-Wesley, Reading, MA.
  • [91] Schneider, R. (1993). Convex Bodies: The Brunn-Minkowski Theory. Cambridge Univ. Press.
  • [92] Shafie, Kh., Worsley, K. J., Wolforth, M. and Evans A. C. (1998). Rotation space: detecting functional activation by searching over rotated and scaled filters. NeuroImage 7 S755.
  • [93] Shafie, Kh. (1998). Ph.D. dissertation, Dept. Math. Statist., McGill Univ.
  • [94] Siegmund, D. (1982). Large deviations for boundary crossing probabilities. Ann. Probab. 10 581-588.
  • [95] Siegmund, D. (1986). Boundary crossing probabilities and statistical applications. Ann. Statist. 14 361-404.
  • [96] Siegmund, D. (1988). Approximate tail probabilities for the maxima of some random fields. Ann. Probab. 16 487-501.
  • [97] Siegmund, D. (1992). Tail approximations for maxima of random fields. In Probability Theory. 147-158. de Gruyter, Berlin.
  • [98] Siegmund, D. O. and Worsley, K. J. (1995). Testing for a signal with unknown location and scale in a stationary Gaussian random field. Ann. Statist. 23 608-639.
  • [99] Siegmund, D. and Zhang, H. (1993). The expected number of local maxima of a random field and the volume of tubes. Ann. Statist. 21 1948-1966.
  • [100] Slepian, D. (1962). The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 42 463-501.
  • [101] Slepian, D. (1963). On the zeroes of Gaussian noise. In Time Series Analysis (M. Rosenblatt, ed.) 104-115. Wiley, New York.
  • [102] Smoot, G. F., Bennett, C. L., Kogut, A., Wright, E. L., Aymon, J., Boggess, N. W., Cheng, E. S., De Amici, G., Gulkis, S., Hauser, M. G., Hinshaw, G., Jackson, P. D., Janssen,
  • R. F., Tenorio, L., Weiss, R. and Wilkinson, D. T. (1992). Structure in the COBE differential microwave radiometer first-year maps. Astrophys. J. 396 L1-L5.
  • [103] Spivak, M. (1979). A Comprehensive Introduction to Differential Geometry 1, 2nd ed. Publish or Perish, Wilmington, DE.
  • [104] Sudakov, N. V. (1969). Gaussian and Cauchy measures and -entropy. Soviet Math. Dokl. 10 310-313.
  • [105] Sudakov, N. V. (1971). Gaussian random processes, and measures of solid angles in Hilbert space. Soviet Math. Dokl. 12 412-415.
  • [106] Sun, J. (1991). Significance levels in exploratory projection pursuit. Biometrika 78 759-769.
  • [107] Sun, J. (1993). Tail probabilities of the maxima of Gaussian random fields. Ann. Probab. 21 34-71.
  • [108] Sun, J. (1993). Some practical aspects of exploratory projection pursuit. SIAM J. Sci. Comput. 14 68-80.
  • [109] Sun, J. (1997). Personal communication.
  • [110] Talagrand, M. (1987). Regularity of Gaussian processes. Acta Math. 159 99-149.
  • [111] Talagrand, M. (1992). A simple proof of the majorizing measure theorem. Geom. Funct. Anal. 2 118-125.
  • [112] Talagrand, M. (1994). Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22 28-76.
  • [113] Talagrand, M. (1994). The supremum of some canonical processes. Amer. J. Math. 116 283-325.
  • [114] Tomita, H. (1990). In Formation, Dynamics and Statistics of Patterns (K. Kawasaki, M. Suzuki and A. Onuki, eds.) 1 113-157. World Scientific, Singapore.
  • [115] Turmon, M. (1995). Assessing generalization of feedforward neural networks. Ph.D. dissertation, Cornell Univ.
  • [116] Wallace, A. H. (1968). Differential Topology: First Steps. Benjamin, New York.
  • [117] Weyl, H. (1939). On the volume of tubes. Amer. J. Math. 61 461-472.
  • [118] Woodroofe, M. and Takahashi, H. (1982). Asymptotic expansions for the error probabilities of some repeated significance tests. Ann. Statist. 10 895-908. [Correction: (1985) 13 837.]
  • [119] Worsley, K. J. (1994). Local maxima and the expected Euler characteristic of excursion sets of 2 F and t fields. Adv. in Appl. Probab. 26 13-42.
  • [120] Worsley, K. J. (1995). Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images. Ann. Statist. 23 640-669.
  • [121] Worsley, K. J. (1995). Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics. Adv. in Appl. Probab. 27 943-959.
  • [122] Worsley, K. J. (1997). The geometry of random images. Chance 9 27-40.
  • [123] Worsley, K. J. (1998). Testing for signals with unknown location and scale in a 2 random field, with an application to fMRI. Preprint.
  • [124] Ylvisaker, N. D. (1965). The expected number of zeros of a stationary Gaussian process. Ann. Math. Statist. 36 1043-1046.
  • [125] Zaanen, A. C. (1956). Linear Analysis. North-Holland, Amsterdam.