Probability Theory and Related Fields

, Volume 87, Issue 4, pp 459–467

Asymptotic behaviour of densities of stable semigroups of measures

  • Jacek Dziubański

DOI: 10.1007/BF01304275

Cite this article as:
Dziubański, J. Probab. Th. Rel. Fields (1991) 87: 459. doi:10.1007/BF01304275


We prove that densities of the measures in a strictly stable semigroup (ht) of symmetric measures on a homogeneous group, if they exist, have the following asymptotic behaviour:
$$\mathop {\lim |}\limits_{|x| \to \infty } x|^{Q + \alpha } \cdot h_1 (x) = k(\bar x),$$
where α is the characteristic exponent,\(\bar x = |x|^{ - 1} x\), andk is the density of the Lévy measure associated to the semigroup. Moreover, if\(k(\bar x) = 0\) a more precise description is given.

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Jacek Dziubański
    • 1
  1. 1.Institute of Mathematics Polish Academy of SciencesMathematical Institute University of WrocławWrocławPoland