Journal of Mathematical Sciences

, Volume 238, Issue 4, pp 406–414 | Cite as

On an Exponential Functional for Gaussian Processes and Its Geometric Foundations

  • R. A. VitaleEmail author

After setting geometric notions, we revisit an exponential functional which has arisen in several contexts, with special attention to a set of geometric parameters and associated inequalities.


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  1. 1.
    C. Borell, “On a certain exponential inequality for Gaussian processes,” Extremes, 9, 169–176 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    S. Chevet, “Processus gaussiens et volumes mixtes,” Z. Wahrsch. Verw. Gebiete, 36, 47–65 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    R. M. Dudley, “The sizes of compact subsets of Hilbert space and continuity of Gaussian processes,” J. Functional Analysis, 1, 290–330 (1967).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    X. Fernique, “Corps convexes et processus gaussiens de petit rang,” Z. Wahrsch. Verw. Gebiete, 35, 349–353 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    L. Gurvits, “A short proof, based on mixed volumes, of Liggett’s theorem on the convolution of ultra-logconcave sequences,” Electron. J. Combin., 16, Note 5 (2009).Google Scholar
  6. 6.
    H. Hadwiger, Vorlesungen über Inhalt, Oberflache, und Isoperimetrie, Springer Verlag, Berlin (1957).CrossRefzbMATHGoogle Scholar
  7. 7.
    H. Hadwiger, “Das Wills’sche Funktional,” Monatsh. Math., 79, 213–221 (1975).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    H. Hadwiger, “Gitterpunktanzahl im Simplex und Willssche Vermutung,” Math. Ann., 239, 271–288 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    K. Itô and M. Nisio, “On the oscillation functions of Gaussian processes,” Math. Scand., 22, 209–223, 1968 (1969).Google Scholar
  10. 10.
    D. A. Klain, “A short proof of Hadwiger’s characterization theorem,” Mathematika, 42, 329–339 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    H. Le, “On bounded Gaussian processes,” Statist. Probab. Lett., 78, 669–674 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M. Ledoux, The Concentration of Measure Phenomenon, Amer. Math. Soc., Providence (2001).zbMATHGoogle Scholar
  13. 13.
    M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer-Verlag, New York (1991).CrossRefzbMATHGoogle Scholar
  14. 14.
    M. Lifshits, Gaussian Random Functions, Kluwer, Boston (1995).CrossRefzbMATHGoogle Scholar
  15. 15.
    T. M. Liggett, “Ultra logconcave sequences and negative dependence,” J. Combin. Theory Ser. A, 79, 315–325 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    P. McMullen, “Non-linear angle-sum relations for polyhedral cones and polytopes,” Math. Proc. Cambridge Philos. Soc., 78, 247–261 (1975).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    P. McMullen, “Inequalities between intrinsic volumes,” Monatsh. Math., 111, 47–53 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    R. Pemantle, “Towards a theory of negative dependence. Probabilistic techniques in equilibrium and nonequilibrium statistical physics,” J. Math. Phys., 41, 1371–1390 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    R. Schneider, Convex Bodies: the Brunn–Minkowski Theory, 2nd ed., Cambridge Univ. Press, New York (2014).zbMATHGoogle Scholar
  20. 20.
    G. C. Shephard, “Inequalities between mixed volumes of convex sets,” Mathematika, 7, 125–138 (1960).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    V. N. Sudakov, “Gaussian random processes and the measures of solid angles in Hilbert space,” Dokl. Akad. Nauk SSSR, 197, 43–45 (1971).MathSciNetGoogle Scholar
  22. 22.
    V. N. Sudakov, “Geometric problems of the theory of infinite-dimensional probability distributions,” Trudy Mat. Inst. Steklov, 141 (1976).Google Scholar
  23. 23.
    V. N. Sudakov, “Geometric problems in the theory of infinite-dimensional probability distributions,” Cover to cover translation of Trudy Mat. Inst. Steklov, 141 (1976), Proc. Steklov Inst. Math., No. 2, 1–178 (1979).Google Scholar
  24. 24.
    B. S. Tsirel’son, “A geometric approach to maximum likelihood estimation for infinite-dimensional Gaussian location. I,” Theory Prob. Appl., 27, 411–418 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    B. S. Tsirel’son, “A geometric approach to maximum likelihood estimation for infinite-dimensional Gaussian location. II,” Theory Prob. Appl., 30, 820–828 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    B. S. Tsirel’son, “A geometric approach to maximum likelihood estimation for infinite-dimensional location. III,” Theory Prob. Appl., 31, 470–483 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    R. A. Vitale, “The Wills functional and Gaussian processes,” Ann. Probab., 24, 2172–2178 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    R. A. Vitale, “A log-concavity proof for a Gaussian exponential bound,” in: T.P. Hill and C. Houdré (eds.) Contemporary Math.: Advances in Stochastic Inequalities, 234, Amer. Math. Soc. (1999), pp. 209–212.Google Scholar
  29. 29.
    R. A. Vitale, “Intrinsic volumes and Gaussian processes,” Adv. Appl. Prob., 33, 354–364 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    R. A. Vitale, “A question of geometry and probability,” in: A Festschrift for Herman Rubin, IMS Lecture Notes Monogr. Ser., 45 (2004), pp. 337–341.Google Scholar
  31. 31.
    R. A. Vitale, “On the Gaussian representation of intrinsic volumes,” Statist. Probab. Lett. 78, 1246–1249 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    J. M. Wills, “Zur Gitterpunktanzahl konvexer Mengen,” Elemente der Math., 28, 57–63 (1973).MathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of ConnecticutStorrsUSA

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