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Large deviations for sums indexed by the generations of a Galton–Watson process
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  • Published: 21 July 2007

Large deviations for sums indexed by the generations of a Galton–Watson process

  • Klaus Fleischmann1 &
  • Vitali Wachtel2 

Probability Theory and Related Fields volume 141, pages 445–470 (2008)Cite this article

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  • 14 Citations

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Abstract

In this paper, we study the large deviation behavior of sums \(S_{{Z}_{n}}\) of i.i.d. random variables X i , where Z n is the nth generation of a supercritical Galton–Watson process. We assume the finiteness of the moments \(EX_{1}^{2}\) and EZ 1 logZ 1 . The underlying interplay of large deviation probabilities of partial sums of the X i and of lower deviation probabilities of Z is clarified. Here, we heavily use lower deviation probability results on Z we recently published in [7].

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Author information

Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117, Berlin, Germany

    Klaus Fleischmann

  2. Technische Universität München Zentrum Mathematik, Bereich M5, 85747, Garching bei München, Germany

    Vitali Wachtel

Authors
  1. Klaus Fleischmann
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  2. Vitali Wachtel
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Corresponding author

Correspondence to Klaus Fleischmann.

Additional information

Supported by the German Science Foundation.

This paper has been written during the time the second author was a staff member of the WIAS Berlin.

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Cite this article

Fleischmann, K., Wachtel, V. Large deviations for sums indexed by the generations of a Galton–Watson process. Probab. Theory Relat. Fields 141, 445–470 (2008). https://doi.org/10.1007/s00440-007-0090-1

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  • Received: 11 August 2006

  • Revised: 05 June 2007

  • Published: 21 July 2007

  • Issue Date: July 2008

  • DOI: https://doi.org/10.1007/s00440-007-0090-1

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Keywords

  • Large deviation probabilities
  • Supercritical Galton–Watson processes
  • Lower deviation probabilities
  • Schröder case
  • Böttcher case
  • Lotka–Nagaev estimator

Mathematics Subject Classification (2000)

  • Primary: 60J80
  • Secondary: 60F10
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