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Lindeberg’s Method for Moderate Deviations and Random Summation

Author

Listed:
  • Peter Eichelsbacher

    (Ruhr-Universität Bochum)

  • Matthias Löwe

    (Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik)

Abstract

We apply Lindeberg’s method, invented to prove a central limit theorem, to analyze the moderate deviations around such a central limit theorem. In particular, we will show moderate deviation principles for martingales as well as for random sums, in the latter situation in both the cases when the limit distribution is Gaussian or non-Gaussian. Moreover, in the Gaussian case we show moderate deviations for random sums using bounds on cumulants, alternatively. Finally, we also prove a large deviation principle for certain random sums.

Suggested Citation

  • Peter Eichelsbacher & Matthias Löwe, 2019. "Lindeberg’s Method for Moderate Deviations and Random Summation," Journal of Theoretical Probability, Springer, vol. 32(2), pages 872-897, June.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:2:d:10.1007_s10959-019-00881-5
    DOI: 10.1007/s10959-019-00881-5
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    References listed on IDEAS

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    1. Huang, Chunmao & Liu, Quansheng, 2012. "Moments, moderate and large deviations for a branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 522-545.
    2. Hanna Döring & Peter Eichelsbacher, 2013. "Moderate Deviations via Cumulants," Journal of Theoretical Probability, Springer, vol. 26(2), pages 360-385, June.
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