Electronic Journal of Probability

Integrability of Seminorms

Andreas Basse-O'Connor

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We study integrability and equivalence of $L^p$<sup></sup>-norms of polynomial chaos elements. Relying on known results for Banach space valued polynomials, we extend and unify integrability for seminorms results to random elements that are not necessarily limits of Banach space valued polynomials. This enables us to prove integrability results for a large class of seminorms of stochastic processes and to answer, partially, a question raised by C. Borell (1979, Seminaire de Probabilites, XIII, 1--3).

Article information

Electron. J. Probab., Volume 16 (2011), paper no. 7, 216-229.

Accepted: 12 January 2011
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G17: Sample path properties
Secondary: 60B11: Probability theory on linear topological spaces [See also 28C20] 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60E15: Inequalities; stochastic orderings

integrability chaos processes seminorms regularly varying distributions

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Basse-O'Connor, Andreas. Integrability of Seminorms. Electron. J. Probab. 16 (2011), paper no. 7, 216--229. doi:10.1214/EJP.v16-853. https://projecteuclid.org/euclid.ejp/1464820176

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  • M. A. Arcones and E. Giné. On decoupling, series expansions, and tail behavior of chaos processes. J. Theoret. Probab. 6 (1993), 101–122.
  • A. Basse and J. Pedersen. Lévy driving moving averages and semimartingales. Stochastic Process. Appl. 119 (2009), 2970–2991.
  • A. Basse-O'Connor and S.-E. Graversen. Path and semimartingale properties of chaos processes. Stochastic Process. Appl. 120 (2010), 522–540.
  • C. Borell. Tail probabilities in Gauss space. In Vector space measures and applications (Proc. Conf., Univ. Dublin, Dublin, 1977), II, Lecture Notes in Phys. 77 (1978), 73–82. Berlin: Springer.
  • C. Borell. On the integrability of Banach space valued Walsh polynomials. In Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78), Lecture Notes in Math. 721 (1979), 1–3. Berlin: Springer.
  • C. Borell. On polynomial chaos and integrability. Probab. Math. Statist. 3 (1984), 191–203.
  • A. de Acosta. Stable measures and seminorms. Ann. Probability 3 (1975), 865–875.
  • V. H. de la Peña and E. Giné. Decoupling. Probability and its Applications (New York), (1999). Springer-Verlag, New York.
  • X. Fernique. Intégrabilité des vecteurs gaussiens. (French) C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1698–A1699.
  • X. Fernique. Fonctions aléatoires gaussiennes, vecteurs aléatoires gaussiens Montreal, QC: Universitée de Montréal Centre de Recherches Matématiques, (1997).
  • M. C. Gemignani. Elementary topology. Corrected reprint of the second edition. Dover Publications, Inc., New York, 1990.
  • J. Hoffmann-Jørgensen. Integrability of seminorms, the 0-1 law and the affine kernel for product measures. Studia Math. 61 (1977), 137–159.
  • J. Hoffmann-Jørgensen. Probability with a view toward statistics. Vol. I. Chapman & Hall Probability Series. New York: Chapman & Hall, (1994).
  • N. C. Jain and D. Monrad. Gaussian quasimartingales. Z. Wahrsch. Verw. Gebiete 59 (1982), 139–159.
  • N. C. Jain and D. Monrad. Gaussian measures in Bp. Ann. Probab. 11 (1983), 46–57.
  • W. Krakowiak and J. Szulga. Random multilinear forms. Ann. Probab. 14 (1986), 955–973.
  • W. Krakowiak and J. Szulga. A multiple stochastic integral with respect to a strictly p-stable random measure. Ann. Probab. 16 (1988), 764–777.
  • S. Kwapień and W. A. Woyczyński. Random series and stochastic integrals: single and multiple. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, (1992)
  • M. Ledoux and M. Talagrand. Probability in Banach spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23, (1991). Springer-Verlag, Berlin. Isoperimetry and processes.
  • M. Marcus and J. Rosiński. Sufficient conditions for boundedness of moving average processes. In Stochastic inequalities and applications, Progr. Probab. 56, (2003), 113–128. Birkhäuser, Basel.
  • D. Nualart. The Malliavin calculus and related topics. (Second edition). Probability and its Applications (New York), (2006). Springer-Verlag, Berlin.
  • G. Pisier. Les inégalités de Khintchine-Kahane, d'après C. Borell. (French) In Séminaire sur la Géométrie des Espaces de Banach (1977–1978), Exp. No. 7, 14 pp, (1978). École Polytech., Palaiseau,.
  • G. Pólya and G. Szegö. Aufgaben und Lehrsätze aus der Analysis. Zweiter Band. Funktionentheorie, Nullstellen, Polynome Determinanten, Zahlentheorie. (German) Vierte Auflage. Heidelberger Taschenbücher, Band 74, (1954). Springer-Verlag, Berlin-Göttingen-Heidelberg.
  • B. Rajput and J. Rosiński. Spectral representations of infinitely divisible processes. Probab. Theory Related Fields 82 (1989), 451–487.
  • J. Rosiński. On stochastic integral representation of stable processes with sample paths in Banach spaces. J. Multivariate Anal. 20 (1986), 277–302.
  • J. Rosiński and G. Samorodnitsky. Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Probab. 21 (1993), 996–1014.
  • J. Rosiński and G. Samorodnitsky. Symmetrization and concentration inequalities for multilinear forms with applications to zero-one laws for Lévy chaos. Ann. Probab. 24 (1996), 422–437.
  • Rudin, Walter. Functional analysis. Second edition. International Series in Pure and Applied Mathematics, (1991). McGraw-Hill, Inc., New York.
  • C. Stricker. Semimartingales gaussiennes–-application au problème de l'innovation. (French) [Gaussian semimartingales–-application to the innovation problem] Z. Wahrsch. Verw. Gebiete 64 (1983), 303–312.