# Lindeberg's method for moderate deviations and random summation

## TL;DR

This paper extends Lindeberg's method to analyze moderate deviations and large deviations for martingales and random sums, providing new principles and bounds for both Gaussian and non-Gaussian cases.

## Contribution

It introduces moderate deviation principles for martingales and random sums, including non-Gaussian cases, and applies cumulant bounds for Gaussian scenarios.

## Key findings

- Moderate deviation principles established for martingales.
- Moderate deviations for random sums with Gaussian and non-Gaussian limits.
- A large deviation principle for certain random sums.

## 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 both, in the case 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.