Skip to main content
SpringerLink
Log in

Distributions of Stable Processes on Spaces of Measurable Functions

  • Chapter
Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference

Part of the book series: Progress in Probability ((PRPR,volume 30))

  • 605 Accesses

Abstract

Let F = F(T,m) be a Banach function space of measurable functions on a σ finite measure space (T,m) and let K:F → L α be a kernel (an integral) operator into the space L α = L α(U, ω) of α-power integrable functions defined on another σ-finite measure space (U,ω). We consider a cylindrical probability μ defined by the characteristic function

$$\hat{\mu}(f)=exp{-\int_{U}|Kf(u)|^{\alpha}w(du)}, \forall f\epsilon IF$$

, for 0 < α < 2. This is the characteristic function of the probability distribution induced by a symmetric α-stable process given by an integral representation. We partly generalize integrability results of these processes by extending the cylindrical probability μ to a countably additive probability.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Buhvalov, A. V. (1974). The integral representation of linear operators. Zap. Naučn. Sem. Leningr. Otdel. Math. Inst. Steklov (LOMI) 47, 5–14.

    MathSciNet  Google Scholar 

  • Cambanis, S. and G. Miller (1981). Linear problems in p-th order and stable processes. SIAM J. Appl. Math. 41, 43–69.

    Article  MathSciNet  MATH  Google Scholar 

  • Cambanis, S., J. Rosiński, and W. Woyczynski (1985). Convergence of quadratic forms in p-stable random variables and Θp-radonifying operators. Ann. Probab. 13, 885–897.

    Article  MathSciNet  MATH  Google Scholar 

  • Chevet, S. (1981). Kernel associated with a cylindrical measure. Lect. Notes in Math. 860, 51–84.

    Article  MathSciNet  Google Scholar 

  • Dudley, R.M. (1967). The sizes of compact subsets of Hilbert space and continuity of Gaussian process. J. Funct. Anal. 1, 290–330.

    Article  MathSciNet  MATH  Google Scholar 

  • Dunford, N., and J. T. Schwartz (1958). Linear Operators. Part I. Interscience, New York.

    Google Scholar 

  • Federer, H. (1969). Geometric measure theory. Springer-Verlag New York Inc., New York.

    MATH  Google Scholar 

  • Gorgadze, Z. G., V. I. Tarieladze, and S. A. Chobanyan (1978). Gaussian covariances in Banach sublattices of the space L 0(T,∑,v). Soviet Math. Dokl. 19, 885–888.

    MATH  Google Scholar 

  • Gretsky, N. E. and J. J. Uhl, Jr. (1981). Carleman and Korotkov operators on Banach spaces. Acta Sci. Math. 43, 207–218.

    MathSciNet  MATH  Google Scholar 

  • Halmos, P. R. and V. S. Sunder (1978). Bounded Integral Operators on L 2-spaces. Springer-Verlag, Berlin-Heidelberg-New York.

    Google Scholar 

  • Korotkov, V. B. (1974). Integral representation of linear operators. Sibirsk. Mat. Ž. 15, 528–545 (in Russian).

    Google Scholar 

  • Kwapień, S. (1970). On a theorem of L. Schwartz and its applications to absolutely summing operators. Studia Math. 38,193–201.

    MathSciNet  MATH  Google Scholar 

  • Linde, W. (1983). Infinitely Divisible and Stable Measures on Banach Spaces. Teubner-Texte zur Math., 58, BSB B.G. Teubner Verlagsesellschaft, Leipzig.

    Google Scholar 

  • Linde, W., V. Mandrekar, and A. Weron (1980). p-stable measures and p-absolutely summing operators. Lect. Notes in Math. 828, 167–177.

    Article  MathSciNet  Google Scholar 

  • McConnell, T. R. (1985). Stable-bounded subsets of Lα, and sample unboundedness of symmetric stable processes. J. Funct. Anal. 60, 265–279.

    Article  MathSciNet  MATH  Google Scholar 

  • Norvaiša, R. (1991). Poisson and Brownian motion random functions with applications. Lith. Acad. of Sc., Inst., of Math. and Inf., Preprint No 13, pp. 58.

    Google Scholar 

  • Persson, A. (1969). On some properties of p-nuclear and p-integral operators. Studia Math. 33, 213–222.

    MathSciNet  MATH  Google Scholar 

  • Rosiński, J. (1980). Remarks on Banach spaces of stable type. Probab. and Math. Stat. 1, 67–71.

    MATH  Google Scholar 

  • Rosiński, J. (1986). On stochastic integral representation of stable processes with sample paths in Banach spaces. J. Mult. Anal. 20, 277–307.

    Article  MATH  Google Scholar 

  • Samorodnitsky, G. (1990). Integrability of stable processes. Center for Stoch. Proc., Technical Report No. 301.

    Google Scholar 

  • Schachermayer, W. (1981). Integral operator on L p spaces. Indiana Univ. Math. J. 30, 123–140.

    Article  MathSciNet  MATH  Google Scholar 

  • Schaefer, H. H. (1980). Topological Vector Spaces. Springer-Verlag, New York-Heidelberg-Berlin.

    MATH  Google Scholar 

  • Schep, A. R. (1979). Kernel operators. Proc. Acad. Amsterdam A 82, 39–53 (Ind. Math. 41).

    MathSciNet  Google Scholar 

  • Schep, A. R. (1980). Generalized Carleman operators. Proc. Acad. Amsterdam A 83, 49–59 (Ind. Math. 42).

    MathSciNet  Google Scholar 

  • Schep, A. R. (1981). Compactness properties of an operator which imply that it is an integral operator. Trans. Amer. Math. Soc. 265, 111–119.

    Article  MathSciNet  MATH  Google Scholar 

  • Suchanecki, Z. (1987). Linear kernels of p-stable measures. Probab. and Math. Stat. 8, 77–80.

    MathSciNet  MATH  Google Scholar 

  • Tarieladze, V. I. (1991). On Gaussian covariances in Banach lattices of measurable functions. Manuscript.

    Google Scholar 

  • Vakhania, N. N., V. I. Tarieladze, and S. A. Chobanyan (1985). Probability Distributions on Banach Spaces. Nauka, Moscow (in Russian) [Engl. trans.: Mathematics and Its Applications (Soviet Series), Reidel, Dordrecht, 1987].

    Google Scholar 

  • Zaanen, A. C. (1983). Riesz Spaces. II. North-Holland Publ. Company, Netherlands.

    MATH  Google Scholar 

  • Zinn, J. (1976). Admissible translates of stable measures. Studia Math. 54, 245–257.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics Lithuanian, Academy of Sciences, Akademijos 4, Vilnius, 2600, Lithuania

    Rimas Norvaiša

Authors
  1. Rimas Norvaiša
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. of Mathematics, MIT, Cambridge, MA, 02139, USA

    Richard M. Dudley

  2. Department of Mathematics, Tufts University, Medford, MA, 02178, USA

    Marjorie G. Hahn

  3. Dept. of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    James Kuelbs

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer Science+Business Media New York

About this chapter

Cite this chapter

Norvaiša, R. (1992). Distributions of Stable Processes on Spaces of Measurable Functions. In: Dudley, R.M., Hahn, M.G., Kuelbs, J. (eds) Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference. Progress in Probability, vol 30. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0367-4_12

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0367-4_12

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6728-7

  • Online ISBN: 978-1-4612-0367-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation