A note on the distributions of the maximum of linear Bernoulli processes

Sergey G Bobkov (University of Minnesota)

Abstract


We give a characterization of the family of all probability measures on the extended line $(-\infty,+\infty]$, which may be obtained as the distribution of the maximum of some linear Bernoulli process.

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Pages: 266-271

Publication Date: May 25, 2008

DOI: 10.1214/ECP.v13-1375

References

  1. Alfsen, Erik M. Compact convex sets and boundary integrals.Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57.Springer-Verlag, New York-Heidelberg, 1971. x+210 pp. MR0445271 (56 #3615)
  2. Billingsley, Patrick. Probability and measure.Wiley Series in Probability and Mathematical Statistics.John Wiley & Sons, New York-Chichester-Brisbane, 1979. xiv+515 pp. ISBN: 0-471-03173-9 MR0534323 (80h:60001)
  3. Borell, Christer. Convex measures on locally convex spaces. Ark. Mat. 12 (1974), 239--252. MR0388475 (52 #9311)
  4. Borell, Christer. The Ehrhard inequality. C. R. Math. Acad. Sci. Paris 337 (2003), no. 10, 663--666. MR2030108 (2004k:60102)
  5. Cirel'son, B. S. Density of the distribution of the maximum of a Gaussian process.(Russian) Teor. Verojatnost. i Primenen. 20 (1975), no. 4, 865--873. MR0394834 (52 #15633)
  6. Ehrhard, Antoine. Symétrisation dans l'espace de Gauss.(French) [Symmetrization in Gaussian spaces] Math. Scand. 53 (1983), no. 2, 281--301. MR0745081 (85f:60058)
  7. Hervé, Michel. Sur les représentations intégrales à l'aide des points extrémaux dans un ensemble compact convexe métrisable.(French) C. R. Acad. Sci. Paris 253 1961 366--368. MR0143010 (26 #577)
  8. Hoffmann-Jørgensen, J.; Shepp, L. A.; Dudley, R. M. On the lower tail of Gaussian seminorms. Ann. Probab. 7 (1979), no. 2, 319--342. MR0525057 (80j:60051)
  9. Kantorovich, L. V.; Akilov, G. P. Functional analysis.Translated from the Russian by Howard L. Silcock.Second edition.Pergamon Press, Oxford-Elmsford, N.Y., 1982. xiv+589 pp. ISBN: 0-08-023036-9; 0-08-026486-7 MR0664597 (83h:46002)
  10. Meyer, Paul-A. Probability and potentials.Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London 1966 xiii+266 pp. MR0205288 (34 #5119)
  11. Prékopa, András. Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. (Szeged) 32 (1971), 301--316. MR0315079 (47 #3628)


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