Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Exponential integrability of sub-Gaussian vectors
Download PDF
Download PDF
  • Published: December 1990

Exponential integrability of sub-Gaussian vectors

  • Ryoji Fukuda1 

Probability Theory and Related Fields volume 85, pages 505–521 (1990)Cite this article

  • 283 Accesses

  • 12 Citations

  • Metrics details

Summary

In this paper we define two classes of Banach space (B, ∥·∥)-valued random vectors called sub-Gaussian vectors and γ-sub-Gaussian vectors. The main purpose of this paper is to prove the exponential integrability of a sub-Gaussian vectorX, that is,\(\mathbb{E}[e^{\varepsilon \parallel X\parallel 2} ]< \infty\) for some ε>0, in the case whereB=L p . On the other hand, using the arguments ofX. Fernique and M. Talagrand, we also show that the exponential integrability of a γ-sub-Gaussian vector in an arbitrary separable Banach space.

These two definitions of sub-Gaussian vectors and γ-sub-Gaussian vectors are not comparable, and neither of these definitions is a necessary condition for the exponential integrability. We shall give illuminating examples.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Buldygin, V.V., Kozachenko, Yu.V.: Sub-Gaussian random variables. Ukrainan Math. J.32, 483–489 (1980)

    Google Scholar 

  2. Fernique, X.: Régularité des trajectoires des fonctions aléatoires gaussiennes. (Lect. Notes Math., vol. 480) Berlin Heidelberg New York: Springer 1974

    Google Scholar 

  3. Heinkel, B.: Mesures majorantes et théorème de la limite centrale dansC(S) Z. Wahrscheinlichkeitstheor. Verw. Geb.38, 339–351 (1977)

    Google Scholar 

  4. Ito, K., Nisio, M.: On the convergence of sums of independent Banach space valued random variables. Nagoya Math. J.47, 15–28 (1968)

    Google Scholar 

  5. Kahane, J.P.: Some random series of functions. 2nd ed.: Cambridge: Cambridge University Press 1985

    Google Scholar 

  6. Kahane, J.P.: Propriétés locales des fonctions à séries de Fourier aléatoires. Studia Math.19, 1–25 (1960)

    Google Scholar 

  7. Kadec, M.I., Pelczyński, A.: Bases, lacunary sequences and complemented subspaces in the spaceL. Studia Math.21 (1962)

  8. Kuo, H.H.: Gaussian measures in Banach spaces. (Lect. Notes Math., vol. 463) Berlin Heidelberg New York: Springer 1975

    Google Scholar 

  9. Kwapién, S.: Probability in Banach space. Oberwolfach (1975). (Lect. Notes, Math., vol. 526, pp. 157–158) Berlin Heidelberg New York: Springer 1976

    Google Scholar 

  10. Jain, N.C., Marcus, M.B.: Continuity of subgaussian processes. In: Kuelbs, J. (ed) Probability on Banach spaces. vol. 4, pp. 81–196 New York: Dekker 1978

    Google Scholar 

  11. Jain, N.C., Marcus, M.B.: Integrability of infinite sums of independent vector-valued random variables. Trans. Am. Math. Soc.212, 1–36 (1975)

    Google Scholar 

  12. Jain, N.C., Marcus, M.B.: Central limit theorems forC(S)-valued random variables. J. Funct. Anal.19, 216–231 (1975)

    Google Scholar 

  13. Maurey, B., Pisier, G.: Séries de variables aléatoires vectorielles indépendantes at propriétés géométriques des espaces de Banach. Studia Math.58, 45–90 (1976)

    Google Scholar 

  14. Talagrand, M.: Regularity of Gaussian processes. Acta Math.159, 99–149 (1987)

    Google Scholar 

  15. Yosida, K.: Functional analysis (Grundlehren vol. 123) Berlin Heidelberg New York: Springer 1966

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Kyushu University 33, 812, Fukuoka, Japan

    Ryoji Fukuda

Authors
  1. Ryoji Fukuda
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Fukuda, R. Exponential integrability of sub-Gaussian vectors. Probab. Th. Rel. Fields 85, 505–521 (1990). https://doi.org/10.1007/BF01203168

Download citation

  • Received: 23 June 1989

  • Revised: 24 November 1989

  • Issue Date: December 1990

  • DOI: https://doi.org/10.1007/BF01203168

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Banach Space
  • Stochastic Process
  • Probability Theory
  • Random Vector
  • Mathematical Biology
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature