Probability Theory and Related Fields

, Volume 77, Issue 3, pp 325–342 | Cite as

Hypercontraction principle and random multilinear forms

  • Wieslaw Krakowiak
  • Jerzy Szulga


We study a Banach space valued random multilinear forms in independent real random variables extensively using the concept of hypercontractive maps between Lq-spaces. We show that multilinear forms share with linear forms a lot of properties, like comparability of Lq-,L0-and almost sure convergence.


Banach Space Stochastic Process Probability Theory Statistical Theory Linear Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Assouad, P.: Espaces p-lieese et q-conves. Inegalités de Burkholder. Séminaire Maurey-Schwartz, Exp. XV, Ecole Polytechnique Paris 1975Google Scholar
  2. 2.
    Becker, W.: Inequalities in Fourier analysis. Ann. Math. 102, 159–182 (1975)Google Scholar
  3. 3.
    Billingsley, P.: Convergence of probability measures. New York London Sydney Toronto:Wiley 1968Google Scholar
  4. 4.
    Bonami, A.: Etude des coefficients de Fourier des fonctions de L p (B). Ann. Inst. Fourier 20, 335–402 (1970)Google Scholar
  5. 5.
    Borell, C.: On the integrability of Banach space valued Walsh polynomials. Seminaire de Probabilite XIII, Strasbourg 1977/78 (Lect. Notes Math, vol.721, pp. 1–3) Berlin Heidelberg New York: Springer 1979Google Scholar
  6. 6.
    Borell, C.: On polynomial chaos and integrability. Probab. Math. Statist. 3, 191–203 (1984)Google Scholar
  7. 7.
    Cambanis, S., Ropsinski, J., Woyczynski, W.A.: Convergence of quadratic forms in p-stable random variables and θp-radonifying operators. Ann. Probab. 13, 885–897 (1985)Google Scholar
  8. 8.
    Dehling, H., Denker, M., Woyczynski, W.A.: Resampling U-statistics using p-stable laws. Preprint $86-50, Case Western Reserve University 1986Google Scholar
  9. 9.
    Diestel, J., Uhl, J.J. Jr., Vector measures. Providence, R.I.:Am. Math. Soc. 1977Google Scholar
  10. 10.
    Feller, W.: An introduction to probability theory and its applications, vol. II. New York London Sydney:Wiley 1966Google Scholar
  11. 11.
    Gross, L.: Logarithm Sobolev inequalities. Am. J. Math. 97, 1061–1083 (1973)Google Scholar
  12. 12.
    Hoffmann-Jørgensen, J.: Sums of independent Banach space valued random variables. Aarhus University Preprint Series 15, 1–96 (1972/73)Google Scholar
  13. 13.
    Janson, C.: On hypercontractivity for multipliers on orthogonal polynomials. Arkiv. Math. 21, 97–110 (1983)Google Scholar
  14. 14.
    Kahane, J.P.: Some random series of function. Lexington, Mass.: D.C. Health Co. 1968Google Scholar
  15. 15.
    Krakowiak, W., Szulga, J.: Random multilinear forms. Ann. Probab. 14, 955–973 (1986)Google Scholar
  16. 16.
    Krakowiak, W., Szulga, J.: A multiple stochastic integral with respect to a strictly p-stable random measure. Preprint # 86-48, Case Western Reserve University 1986 (to appear in Ann. Probab.)Google Scholar
  17. 17.
    Kwapien, S.: Decoupling inequalities for polynomial chaos. Seminar Notes on Multiple Stochastic Integration, Polynomial Chaoses and their Applications. Case Western Reserve University 1985Google Scholar
  18. 18.
    Lindenstrauss, J., Tzafirir, L.: Classical Banach spaces, vol. II. Berlin Heidelberg New York: Springer 1979Google Scholar
  19. 19.
    Marcinkiewicz, J., Zygmund, A.: Sur les fonctions independentes. Fundamenta Mathematicae 29, 60–90 (1937)Google Scholar
  20. 20.
    Nelson, E.:The free Markov fields. J. Funct. Anal. 12, 211–227 (1973)Google Scholar
  21. 21.
    Paley, R.E.A.C., Zygmund, A.: A note analytic functions on the unit circle. Proc. Cambridge Philos. Soc. 28, 266–272 (1932)Google Scholar
  22. 22.
    Rolewicz, S.: Metric linear spaces. Warszawa:Polish Scientific Publishers 1973Google Scholar
  23. 23.
    Rosinski, J., Woyczynski, W.A.: On Itô stochastic integration with respect to p-stable motion: inner clock, integrability of sample paths, double and multiple integrals. Ann. Probab. 14, 271–286 (1986)Google Scholar
  24. 24.
    Rosinski, J., Woycynski, W.A.: Multilinear forms in Pareto-like random variables and product random measures. Colloquium Mathematicum, S. Hartman Festschrift, 51, 303–313 (1987)Google Scholar
  25. 25.
    Shohat, J.A.: Théorie générale des polynomes orthogonaux de Tchebycheff. Mém. Sci. Math., Fasc. 66, Paris 1934Google Scholar
  26. 26.
    Sjörgen, P.: On the convergence of bilinear and quadratic forms in independent random variables. Stud. Math. 71,285–296 (1982)Google Scholar
  27. 27.
    Surgailis, D.: On the multiple stable integral. Z. Wahrscheinlichkeistheor. Verw. Geb. 70, 621–632 (1985)Google Scholar
  28. 28.
    Szulga, J.: On hypercontractivity of α-stable random variables, 0<α<2. Technical Report # 196, Center for Stochastic Processes, University of North Carolina at Chapel Hill 1987Google Scholar
  29. 29.
    Sztencel, R.: On boundedness and convergence of some Banach space valued random series. Probab. Math. Statist. 2, Fasc. 1, 83–88 (1981)Google Scholar
  30. 30.
    Weissler, F.B.: Logarithm Sobolev inequalities and hypercontractive estimates on the circle. J. Funct. Anal. 37, 218–234 (1980)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Wieslaw Krakowiak
    • 1
  • Jerzy Szulga
    • 1
  1. 1.Institute of MathematicsWroclaw UniversityWroclawPoland

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