Probability Theory and Related Fields

, Volume 87, Issue 4, pp 459-467

First online:

Asymptotic behaviour of densities of stable semigroups of measures

  • Jacek DziubańskiAffiliated withInstitute of Mathematics Polish Academy of Sciences, Mathematical Institute University of Wrocław

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


We prove that densities of the measures in a strictly stable semigroup (h t ) 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.