Validity of the expected Euler characteristic heuristic. (English) Zbl 1083.60031

Authors’ abstract: We study the accuracy of the expexted Euler characteristic approximation to the distribution of the maximum of a smooth, centered, unit variance Gaussian process \(f\). Using a point process representation of the error, valid for arbitrary smooth processes, we show that the error is in general exponentially smaller than any of the terms in the approximation. We also give a lower bound on this exponential rate of decay in terms of the maximal variance of a family of Gaussian processes \(f^x\), derived from the original process \(f\).


60G15 Gaussian processes
60G60 Random fields
53A17 Differential geometric aspects in kinematics
58A05 Differentiable manifolds, foundations
60G17 Sample path properties
62M40 Random fields; image analysis
60G70 Extreme value theory; extremal stochastic processes


Full Text: DOI arXiv


[1] Adler, R. J. (1981). The Geometry of Random Fields . Wiley, Chichester. · Zbl 0478.60059
[2] Adler, R. J. (1990). An Introduction to Continuity , Extrema and Related Topics for General Gaussian Processes . IMS, Hayward, CA. · Zbl 0747.60039
[3] Adler, R. J. (2000). On excursion sets, tube formulae, and maxima of random fields. Ann. Appl. Probab. 10 1–74. · Zbl 1171.60338
[4] Adler, R. J. and Taylor, J. E. (2005). Random Fields and Their Geometry . Birkhäuser, Boston. To appear. Most chapters currently available at ie.technion.ac.il/Adler.phtml. · Zbl 1060.60052
[5] Azaïs, J.-M., Bardet, J.-M. and Wschebor, M. (2002). On the tails of the distribution of the maximum of a smooth stationary Gaussian process. ESAIM Probab. Statist. 6 177–184. · Zbl 1009.60022
[6] Azaïs, J.-M. and Wschebor, M. (2005). On the distribution of the maximum of a Gaussian field with \(d\) parameters. Ann. Appl. Probab. 15 254–278. · Zbl 1079.60031
[7] Brillinger, D. R. (1972). On the number of solutions of systems of random equations. Ann. Math. Statist. 43 534–540. · Zbl 0238.60040
[8] Cao, J. and Worsley, K. J. (1999). The geometry of correlation fields with an application to functional connectivity of the brain. Ann. Appl. Probab. 9 1021–1057. · Zbl 0961.60052
[9] Cao, J. and Worsley, K. J. (1999). The detection of local shape changes via the geometry of Hotelling’s \(T^ 2\) fields. Ann. Statist. 27 925–942. · Zbl 0986.62076
[10] Delmas, C. (1998). An asymptotic expansion for the distribution of the maximum of a class of Gaussian fields. C. R. Acad. Sci. Paris Sér. I Math. 327 393–397. · Zbl 0920.60036
[11] Federer, H. (1959). Curvature measures. Trans. Amer. Math. Soc. 93 418–491. · Zbl 0089.38402
[12] 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. JSTOR: · Zbl 0723.62018
[13] Kratz, M. F. and Rootzén, H. (1997). On the rate of convergence for extremes of mean square differentiable stationary normal processes. J. Appl. Probab. 34 908–923. · Zbl 0903.60043
[14] Kuriki, S., Takemura, A. and Taylor, J. (2004). Asymptotic evaluation of the remainder of tube formula approximation to tail probabilities. Unpublished manuscript.
[15] Kuriki, S. and Taylor, J. (2003). The tube method for Gaussian fields with inhomogeneous mean and/or variance—smooth and piecewise smooth cases. Unpublished manuscript.
[16] Piterbarg, V. I. (1981). Comparison of distribution functions of maxima of Gaussian processes. Theory Probab. Appl. 26 687–705. · Zbl 0488.60051
[17] Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields . Amer. Math. Soc., Providence, RI. (Translated from the Russian by V. V. Piterbarg, revised by the author.) · Zbl 0841.60024
[18] Sun, J. (1993). Tail probabilities of the maxima of Gaussian random fields. Ann. Probab. 21 34–71. JSTOR: · Zbl 0772.60038
[19] Takemura, A. and Kuriki, S. (2004). Some results on geometry of isotropic and spherically isotropic smooth Gaussian fields. Unpublished manuscript. · Zbl 1047.62055
[20] Takemura, A. and Kuriki, S. (2002). On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains. Ann. Appl. Probab. 12 768–796. · Zbl 1016.60042
[21] Takemura, A. and Kuriki, S. (2003). Tail probability via the tube formula when the critical radius is zero. Bernoulli 9 535–558. · Zbl 1063.60078
[22] Taylor, J. and Adler, R. J. (2003). Euler characteristics for Gaussian fields on manifolds. Ann. Probab. 31 533–563. · Zbl 1026.60039
[23] Weyl, H. (1939). On the volume of tubes. Amer. J. Math. 61 461–472. · Zbl 0021.35503
[24] 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. · Zbl 0836.60043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.