Journal of Theoretical Probability

, Volume 32, Issue 2, pp 872–897

# Lindeberg’s Method for Moderate Deviations and Random Summation

• Peter Eichelsbacher
• Matthias Löwe
Article

## 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.

## Keywords

Random sums Moderate and large deviations Lindeberg’s method

## Mathematics Subject Classification

60F05 60F10 60G50

## Notes

### Acknowledgements

We are very grateful to an anonymous referee for a very careful reading of a first version of this manuscript. His comments helped to improve the correctness of the paper.

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