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Convolution equivalence and distributions of random sums
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  • Published: 30 November 2007

Convolution equivalence and distributions of random sums

  • Toshiro Watanabe1 

Probability Theory and Related Fields volume 142, pages 367–397 (2008)Cite this article

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Abstract

A serious gap in the Proof of Pakes’s paper on the convolution equivalence of infinitely divisible distributions on the line is completely closed. It completes the real analytic approach to Sgibnev’s theorem. Then the convolution equivalence of random sums of IID random variables is discussed. Some of the results are applied to random walks and Lévy processes. In particular, results of Bertoin and Doney and of Korshunov on the distribution tail of the supremum of a random walk are improved. Finally, an extension of Rogozin’s theorem is proved.

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Authors and Affiliations

  1. Center for Mathematical Sciences, The University of Aizu, Aizu-Wakamatsu, Japan

    Toshiro Watanabe

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  1. Toshiro Watanabe
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Correspondence to Toshiro Watanabe.

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Cite this article

Watanabe, T. Convolution equivalence and distributions of random sums. Probab. Theory Relat. Fields 142, 367–397 (2008). https://doi.org/10.1007/s00440-007-0109-7

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  • Received: 20 October 2006

  • Revised: 26 September 2007

  • Published: 30 November 2007

  • Issue Date: November 2008

  • DOI: https://doi.org/10.1007/s00440-007-0109-7

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Keywords

  • Convolution equivalence
  • Subexponentiality
  • O-subexponentiality
  • Infinite divisibility
  • Random sum
  • IID

Mathematical Subject Classification (2000)

  • Primary: 60E07
  • 60G50
  • Secondary: 60G51
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