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Asymptotic behaviour of densities of stable semigroups of measures
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  • Published: December 1991

Asymptotic behaviour of densities of stable semigroups of measures

  • Jacek Dziubański1 

Probability Theory and Related Fields volume 87, pages 459–467 (1991)Cite this article

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Summary

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.

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Authors and Affiliations

  1. Institute of Mathematics Polish Academy of Sciences, Mathematical Institute University of Wrocław, Pl. Grunwaldzki 2/4, PL-50-384, Wrocław, Poland

    Jacek Dziubański

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  1. Jacek Dziubański
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Dziubański, J. Asymptotic behaviour of densities of stable semigroups of measures. Probab. Th. Rel. Fields 87, 459–467 (1991). https://doi.org/10.1007/BF01304275

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  • Received: 22 March 1989

  • Revised: 30 June 1990

  • Issue Date: December 1991

  • DOI: https://doi.org/10.1007/BF01304275

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Keywords

  • Stochastic Process
  • Asymptotic Behaviour
  • Probability Theory
  • Mathematical Biology
  • Homogeneous Group
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