Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T04:11:22.788Z Has data issue: false hasContentIssue false
  • English
  • Français

Intrinsic volumes and Gaussian processes

Part of: General convexity Stochastic processes Distribution theory - Probability Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Richard A. Vitale
Richard A. Vitale*
Affiliation:
University of Connecticut
*
∗ Postal address: Department of Statistics, University of Connecticut, Storrs, CT 06269, USA. Email address: rvitale@uconnvm.uconn.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Intrinsic volumes are key functionals in convex geometry and have also appeared in several stochastic settings. Here we relate them to questions of regularity in Gaussian processes with regard to Itô–Nisio oscillation and metrization of GB/GC indexing sets. Various bounds and estimates are presented, and questions for further investigation are suggested. From alternate points of view, much of the discussion can be interpreted in terms of (i) random sets and (ii) properties of (deterministic) infinite-dimensional convex bodies.

Keywords

Convex bodydeviation boundGaussian processGB/GC setintrinsic volumeItô-Nisio oscillationmetricquermassintegralSteiner pointWills functional

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 52A07: Convex sets in topological vector spaces 52A22: Random convex sets and integral geometry 52A39: Mixed volumes and related topics 60D05: Geometric probability and stochastic geometry 60E15: Inequalities; stochastic orderings
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 2 , June 2001 , pp. 354 - 364
Copyright
Copyright © Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Badrikian, A. and Chevet, S. (1974). Mesures cylindriques, espaces de Wiener et fonctions aléatoires gaussiennes (Lecture Notes Math. 379). Springer, Berlin.CrossRefGoogle Scholar
[2] Beer, G. (1993). Topologies on Closed and Closed Convex Sets. Kluwer, Boston.CrossRefGoogle Scholar
[3] Chevet, S. (1973). Épaisseur mixte. C. R. Acad. Sci. Paris A–B 276, A371–374.Google Scholar
[4] Chevet, S. (1976). Processus gaussiens et volumes mixtes. Z. Wahrscheinlichkeitsth. 36, 47–65.CrossRefGoogle Scholar
[5] De la Peña, V. H. and Giné, E. (1999). Decoupling. From Independence to Dependence. Springer, New York.Google Scholar
[6] Dmitrovskii, V. A. (1989). On the integrability of the maximum and the local properties of Gaussian fields. In Probability Theory and Mathematical Statistics, Vol. I, eds Grigelionis, B. et al. Mokslas, Vilnius, pp. 271–284.Google Scholar
[7] Gao, F. and Vitale, R. A. (2001). Intrinsic volumes of the Brownian motion body. To appear in Discrete Comput. Geom. Google Scholar
[8] Groemer, H. (1996). Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge University Press.Google Scholar
[9] Hadwiger, H. (1975). Das Wills'sche Funktional. Monatsh. Math. 79, 213–221.Google Scholar
[10] Hadwiger, H. and Wills, J. M. (1974). Gitterpunktanzahl konvexer Rotationkörper. Math. Ann. 208, 221–232.Google Scholar
[11] Itô, K. and Nisio, M. (1969). On the oscillation functions of Gaussian processes. Math. Scand. 22, 209–223.Google Scholar
[12] Kendall, W. S., van Lieshout, M. N. M. and Baddeley, A. J. (1999). Quermass-interaction processes: conditions for stability. Adv. Appl. Prob. 31, 315–342.Google Scholar
[13] Lifshits, M. A. (1995). Gaussian Random Functions. Kluwer, Boston.Google Scholar
[14] McMullen, P. (1975). Non-linear angle-sum relations for polyhedral cones and polytopes. Math. Proc. Camb. Phil. Soc. 78, 247–261.Google Scholar
[15] McMullen, P. (1991). Inequalities between intrinsic volumes. Monatsh. Math. 111, 47–53.Google Scholar
[16] Sangwine–Yager, J. (1993). Mixed volumes. In Handbook of Convex Geometry, Vol. A, eds Gruber, P. M. and Wills, J. M. North-Holland, New York, pp. 43–71.CrossRefGoogle Scholar
[17] Schneider, R. (1982). Random hyperplanes meeting a convex body. Z. Wahrscheinlichkeitsth. 61, 379–387.Google Scholar
[18] Schneider, R. (1993). Convex Bodies: the Brunn–Minkowski Theory. Cambridge University Press.Google Scholar
[19] Sudakov, V. N. (1971). Gaussian random processes and measures of solid angles in a Hilbert space. Dokl. Akad. Nauk SSSR 197, 43–45 (in Russian). English translation: Soviet Math. Dokl. 12, 412–415.Google Scholar
[20] Sudakov, V. N. (1973). A remark on the criterion of continuity of Gaussian sample functions. In Proc. 2nd Japan–USSR Symp. Prob. Theory (Lecture Notes Math. 330), eds Maruyama, G. and Prokhorov, Yu. V. Springer, Berlin, pp. 444–454.Google Scholar
[21] Sudakov, V. N. (1976). Geometric Problems in the Theory of Infinite-Dimensional Probability Distributions. (Trudy Mat. Inst. Steklov 141). Nauka, Moscow (in Russian). English translation: (1979) American Mathematical Society, Providence, RI.Google Scholar
[22] Tsirelson, B. S. (1982). A geometric approach to maximum likelihood estimation for infinite-dimensional Gaussian location I. Theory Prob. Appl. 27, 411–418.Google Scholar
[23] Tsirelson, B. S. (1985). A geometric approach to maximum likelihood estimation for infinite-dimensional Gaussian location II. Theory Prob. Appl. 30, 820–828.CrossRefGoogle Scholar
[24] Tsirelson, B. S. (1986). A geometric approach to maximum likelihood estimation for infinite-dimensional location III. Theory Prob. Appl. 31, 470–483.Google Scholar
[25] Vitale, R. A. (1985). The Steiner point in infinite dimensions. Israel J. Math. 52, 245–250.Google Scholar
[26] Vitale, R. A. (1993). A class of bounds for convex bodies in Hilbert space. Set-Valued Anal. 1, 89–96.Google Scholar
[27] Vitale, R. A. (1995). On the volume of parallel bodies: a probabilistic derivation of the Steiner formula. Adv. Appl. Prob. 27, 97–101.Google Scholar
[28] Vitale, R. A. (1996). The Wills functional and Gaussian processes. Ann. Prob. 24, 2172–2178.CrossRefGoogle Scholar
[29] Vitale, R. A. (1996). A stochastic argument for the uniqueness of the Steiner point. Rend. Circ. Mat. Palermo Suppl. 41, 241–244.Google Scholar
[30] Vitale, R. A. (1996). Covariance identities for normal variables via convex polytopes. Statist. Prob. Lett. 30, 363–368.Google Scholar
[31] Vitale, R. A. (1999). A log-concavity proof for a Gaussian exponential bound. In Advances in Stochastic Inequalities (Contemp. Math. 234), eds Hill, T. P. and Houdré, C. American Mathematical Society, Providence, RI, pp. 209–212.Google Scholar
[32] Weil, W. (1982). Inner contact probabilities for convex bodies. Adv. Appl. Prob. 14, 582–589.Google Scholar
[33] Weil, W. and Wieacker, J. A. (1984). Densities for stationary random sets and point processes. Adv. Appl. Prob. 16, 324–346.Google Scholar
[34] Wills, J. M. (1973). Zur Gitterpunktanzahl konvexer Mengen. Elemente Math. 28, 57–63.Google Scholar