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Gaussian Random Variables

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Probability in Banach Spaces

Part of the book series: Ergebnisse der Mathematik und ihrer Grenzgebiete ((CLASSICS,volume 23))

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Abstract

With this chapter, we really enter into the subject of Probability in Banach spaces. The study of Gaussian random vectors and processes may indeed be considered as one of the fundamental topics of the theory since it inspires many other parts of the field both in the results themselves and in the techniques of investigation. Historically, the developments also followed this line of progress.

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Notes and Reference

  1. J. Neveu: Processus aléatoires gaussiens. Presses de l’Université de Montréal, 1968

    Google Scholar 

  2. A. Badrikrian, S. Chevet: Mesures cylindriques, espaces de Wiener et fonctions aléatoires gaussiennes. Lecture Notes in Mathematics, vol. 379. Springer, Berlin Heidelberg 1974

    Google Scholar 

  3. X. Fernique: Régularité des trajectoires des fonctions aléatoires gaussiennes. Ecole d’Eté de Probabilités de St-Flour 1974. Lecture Notes in Mathematics, vol. 480. Springer, Berlin Heidelberg 1975, pp. 1–96

    Google Scholar 

  4. H. H. Kuo: Gaussian measures in Banach spaces. Lecture Notes in Mathematics, vol. 436. Springer, Berlin Heidelberg 1975

    Google Scholar 

  5. V. N. Sudakov: Geometric problems of the theory of infinite-dimensional probability distributions. Trudy Mat. Inst. Steklov, vol. 141, 1976

    Google Scholar 

  6. N. C. Jain, M. B. Marcus: Continuity of sub-gaussian processes. Advances in Probability, vol. 4. Dekker, New York 1978, pp. 81–196

    Google Scholar 

  7. X. Fernique: Gaussian random vectors and their reproducing kernel Hilbert spaces. Technical report, University of Ottawa, 1985

    Google Scholar 

  8. N. N. Vakhania, V. I. Tarieladze, S. A. Chobanyan: Probability distributions on Banach spaces. Reidel, Dordrecht 1987

    Book  MATH  Google Scholar 

  9. Ad] R. J. Adler: An introduction to continuity, extrema, and related topics for general Gaussian processes. (1989), notes to appear

    Google Scholar 

  10. C. Borell: Gaussian Radon measures on locally convex spaces. Math. Scand. 38, 265–284 (1976)

    MathSciNet  MATH  Google Scholar 

  11. M. Talagrand: Mesures gaussiennes sur un espace localement convexe. Z. Wahrscheinlichkeitstheor. Verw. Geb. 64, 181–209 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. H. J. Landau, L. A. Shepp: On the supremum of a Gaussian process. Sankhyà A32, 369–378 (1970)

    MathSciNet  MATH  Google Scholar 

  13. X. Fernique: Intégrabilité des vecteurs gaussiens. C. R. Acad. Sci. Paris 270, 1698–1699 (1970)

    MathSciNet  MATH  Google Scholar 

  14. A. V. Skorokod: A note on Gaussian measures in a Banach space. Theor. Probab. Appl. 15, 519–520 (1970)

    Google Scholar 

  15. J. Hoffmann-Jorgensen: Probability in Banach spaces. Ecole d’Eté de Probabilités de St-Flour 1976. Lecture Notes in Mathematics, vol. 598. Springer, Berlin Heidelberg 1976, pp. 1–186

    Google Scholar 

  16. M. B. Marcus, L. A. Shepp: Sample behavior of Gaussian processes. Proc. of the Sixth Berkeley Symposium on Math. Statist. and Probab., vol. 2, 1972, pp. 423–441

    MathSciNet  Google Scholar 

  17. C. Borell: The Brunn-Minskowski inequality in Gauss space. Invent. math. 30, 207–216 (1975)

    Article  MathSciNet  Google Scholar 

  18. A. Ehrhard: Convexité des mesures gaussiennes. Thèse de l’Université de Strasbourg, 1985

    Google Scholar 

  19. M. Talagrand: Sur l’intégrabilité des vecteurs gaussiens. Z. Wahrscheinlichkeitstheor. Verw. Geb. 68, 1–8 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  20. V. Goodman. Characteristics of normal samples. Ann. Probab. 16, 1281–1290 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  21. V. Goodman, J. Kuelbs: Rates of convergence for increments of Brownian motion. J. Theor. Probab. 1, 27–63 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  22. V. Goodman, J. Kuelbs: Rates of convergence for the functional LIL. Ann. Probab. 17, 301–316 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  23. G-K3] V. Goodman, J. Kuelbs: Rates of clustering of some self similar Gaussian processes. (1990), to appear in Probab. Theor. Rel. Fields

    Google Scholar 

  24. R. Azencott: Grandes déviations et applications. Ecole d’Eté de Probabilités de St-Flour 1978. Lecture Notes in Mathematics, vol. 774. Springer, Berlin Heidelberg 1979, pp. 1–176

    Google Scholar 

  25. N. C. Jain: An introduction to large deviations. Probability in Banach Spaces V, Medford (U.S. A.) Lecture Notes in Mathematics, vol. 1153. Springer, Berlin Heidelberg 1985, pp. 273–296

    Google Scholar 

  26. J.-D. Deuschel, D. Stroock: Large deviations. Academic Press, 1989

    Google Scholar 

  27. R. Dudley, J. Hoffmann-Jorgensen, L. A. Shepp: On the lower tail of Gaussian seminorms. Ann. Probab. 7, 319–342 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  28. V. S. Tsirel’son: The density of the maximum of a Gaussian process. Theor. Probab. Appl. 20, 847–856 (1975)

    Article  MATH  Google Scholar 

  29. Y. L. Tong: Probability inequalities in multivariate distributions. Academic Press, New York 1980

    MATH  Google Scholar 

  30. V. N. Sudakov: Gaussian measures, Cauchy measures and e-entropy. Soviet Math. Dokl. 10, 310–313 (1969)

    MATH  Google Scholar 

  31. S. J. Montgomery-Smith: The cotype of operators from C(K). Ph. D. Thesis, Cambridge 1988

    Google Scholar 

  32. S. J. Montgomery-Smith: The distribution of Rademacher sums. Proc. Amer. Math. Soc. 109, 517–522 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  33. V. N. Sudakov: Gaussian random processes and measures of solid angles in Hilbert spaces. Soviet Math. Dokl. 12, 412–415 (1971)

    MathSciNet  MATH  Google Scholar 

  34. V. N. Sudakov: Geometric problems of the theory of infinite-dimensional probability distributions. Trudy Mat. Inst. Steklov, vol. 141, 1976

    Google Scholar 

  35. Y. Gordon: Some inequalities for Gaussian processes and applications. Israel J. Math. 50, 265–289 (1985)

    MATH  Google Scholar 

  36. Y. Gordon: Gaussian processes and almost spherical sections of convex bodies. Ann. Probab. 16, 180–188 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  37. Y. Gordon: Elliptically contoured distributions. Probab. Theor. Rel. Fields 76, 429–438 (1987)

    Article  MATH  Google Scholar 

  38. J.-P. Kahane: Une inégalité du type de Slepian et Gordon sur les processus gaussiens. Israel J. Math. 55, 109–110 (1986)

    MathSciNet  MATH  Google Scholar 

  39. V. N. Sudakov: Gaussian measures, Cauchy measures and e-entropy. Soviet Math. Dokl. 10, 310–313 (1969)

    MATH  Google Scholar 

  40. V. N. Sudakov: A remark on the criterion of continuity of Gaussian sample functions. Proceedings of the Second Japan-USSR Symposium on Probability Theory. Lecture Notes in Mathematics, vol. 330. Springer, Berlin Heidelberg 1973, pp. 444–454

    Book  Google Scholar 

  41. A. Pajor, N. Tomczak-Jaegermann• Subspaces of small codimension of finite dimensional Banach spaces. Proc. Amer. Math. Soc. 97, 637–642 (1986)

    MathSciNet  MATH  Google Scholar 

  42. W. Linde, A. Pietsch: Mappings of Gaussian cylindrical measures in Banach spaces. Theor. Probab. Appl. 19, 445–460 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  43. Ta19] M. Talagrand: A new isoperimetric inequality. (1990), to appear in Geometric Aspects of Functional Analysis, Israel Seminar 1989/90. Lecture Notes in Mathematics. Springer, Berlin Heidelberg

    Google Scholar 

  44. N. Tomczak-Jaegermann: Dualité des nombres d’entropie pour des opérateurs à valeurs dans un espace de Hilbert. C. R. Acad. Sci. Paris 305, 299–301 (1987)

    MathSciNet  MATH  Google Scholar 

  45. B. Carl: Entropy numbers, s-numbers and eigenvalue problems. J. Funct. Anal. 41, 290–306 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  46. S. Chevet: Un résultat sur les mesures gaussiennes. C. R. Acad. Sci. Paris 284, 441–444 (1977)

    MathSciNet  MATH  Google Scholar 

  47. S. Chevet: Séries de variables aléatoires gaussiennes à valeurs dans ELF. Applications aux produits d’espaces de Wiener abstraits Séminaire sur la Géométrie des Espaces de Banach 1977–78. Ecole Polytechnique, Paris 1978

    Google Scholar 

  48. R. Carmona: Tensor product of Gaussian measures. Vector Space Measures and Applications, Dublin 1977. Lecture Notes in Mathematics, vol. 644. Springer, Berlin Heidelberg 1978, pp. 96–124

    Google Scholar 

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Ledoux, M., Talagrand, M. (1991). Gaussian Random Variables. In: Probability in Banach Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20212-4_5

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  • DOI: https://doi.org/10.1007/978-3-642-20212-4_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-20211-7

  • Online ISBN: 978-3-642-20212-4

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