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Séminaire de Probabilités LI pp 239–257Cite as

Large Deviations at the Transition for Sums of Weibull-Like Random Variables

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Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 2301)

Abstract

Deviation probabilities of the sum Sn = X1 + ⋯ + Xn of independent and identically distributed real-valued random variables have been extensively investigated, in particular when X1 is Weibull-like distributed, i.e. \(\log \mathbb {P}(X\geqslant x) \sim -qx^{1-\epsilon }\) as x →∞. For instance, A.V. Nagaev formulated exact asymptotic results for \(\mathbb {P}(S_n>x_n)\) when xn > n1∕2 (see, A.V. Nagaev, 1969). In this paper, we derive rough asymptotic results (at logarithmic scale) with shorter proofs relying on classical tools of large deviation theory and giving an explicit formula for the rate function at the transition xn =  Θ(n1∕(1+𝜖)).

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Acknowledgements

We would like to thank the referee for his comments and his suggestions which pushed us to make some proofs clearer and shorter.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UT3, Toulouse, France

    Fabien Brosset & Pierre Petit

  2. Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, Toulouse, France

    Thierry Klein

  3. ENAC—Ecole Nationale de l’Aviation Civile, Université de Toulouse, Toulouse, France

    Thierry Klein

  4. Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UT2J, Toulouse, France

    Agnès Lagnoux

Authors
  1. Fabien Brosset
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  2. Thierry Klein
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  3. Agnès Lagnoux
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  4. Pierre Petit
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Corresponding author

Correspondence to Agnès Lagnoux .

Editor information

Editors and Affiliations

  1. Laboratoire de Mathématiques de Versailles, Université de Versailles-St-Quentin, Versailles, France

    Prof. Catherine Donati-Martin

  2. Institut Elie Cartan de Lorraine, Vandoeuvre-lès-Nancy, France

    Antoine Lejay

  3. Laboratoire de Mathématiques de Versailles, Université de Versailles-St-Quentin, Versailles, France

    Prof. Alain Rouault

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Brosset, F., Klein, T., Lagnoux, A., Petit, P. (2022). Large Deviations at the Transition for Sums of Weibull-Like Random Variables. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités LI. Lecture Notes in Mathematics(), vol 2301. Springer, Cham. https://doi.org/10.1007/978-3-030-96409-2_8

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