Journal of Theoretical Probability

, Volume 9, Issue 2, pp 533–540 | Cite as

Zeros of the densities of infinitely divisible measures on R n

  • Tomasz Byczkowski
  • Balram S. Rajput
  • Tomasz Żak
Article

Abstract

Let μ be an infinitely divisible probability measure onR n without Gaussian component and let ν be its Lévy measure. Suppose that μ is absolutely continuous with respect to the Lebesgue measure λ. We investigate the structure of the set ℝ n of admissible translates of μ. This yields a unified presentation of previously known results. We also show that ifλ(S)>0 then μ is equivalent to λ, under the assumption that supp μ=R n , whereS is the closure of the semigroup generated by the support of ν.

Key Words

Infinitely divisible measures Poisson measures equivalence with Lebesgue measure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brockett, P. L. (1977). Supports of infinitely divisible neasures on Hilbert space,Ann. Prob. 5, 1012–1017.Google Scholar
  2. 2.
    Brockett, P. L., and Hudson, W. N. (1980). Zeros of the densities of infinitely divisible measures,Ann. Prob. 8, 400–403.Google Scholar
  3. 3.
    Brown, G. (1976), Infinitely divisible distributions and support properties,Proc. Royal Irish Acad., Sect.22A, 227–234.Google Scholar
  4. 4.
    Hewitt, E., and Stromberg, K. (1969).Real and Abstract Analysis, Springer-Verlag, New York.Google Scholar
  5. 5.
    Hudson, W. N., and Mason, J. D. (1975). More on equivalence of infinitely divisible distributions,Ann. Prob. 3, 563–568.Google Scholar
  6. 6.
    Hudson, W. N., and Tucker, H. G. (1975). On admissible translates of infinitely divisible distributions,Z. Wahr. 31, 65–72.Google Scholar
  7. 7.
    Oxtoby, J. C. (1971).Measure and Category, Springer-Verlag, New York.Google Scholar
  8. 8.
    Rajput, B. S. (1977). On the support of symmetric infinitely divisible and stable probability measures on LCTVS,Proc. Amer. Math. Soc. 66, 331–334.Google Scholar
  9. 9.
    Rajput, B. S. (1993). Supports of certain infinitely divisible probability measures on locally convex spaces,Ann. Prob. 21, 886–897.Google Scholar
  10. 10.
    Tortrat, A. (1988). Le support des lois indefiniment divisibles dans un groupe Abelien localement compact,Math. Z. 197, 231–250.Google Scholar
  11. 11.
    Tucker, H. G. (1962). Absolute continuity of infinitely divisible distributions,Pacific J. Math. 12, 1125–1129.Google Scholar
  12. 12.
    Yuan, J. (1979). On supports and equivalence of embeddible probability measures,Bull. Inst. Math. Acad. Sinica 7, 471–478.Google Scholar
  13. 13.
    Yuan, J. (1980). A note on angular semigroups,Semigroup Forum 19, 261–265.Google Scholar
  14. 14.
    Yuan, J. (1983). On the structure of monoids of admissible translates of multivariate probability measures,Semigroup Forum 27, 377–386.Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Tomasz Byczkowski
    • 1
  • Balram S. Rajput
    • 2
  • Tomasz Żak
    • 1
  1. 1.Institute of MathematicsWrocław Technical UniversityWrocław
  2. 2.Departments of MathematicsThe University of TennesseeKnoxville

Personalised recommendations