On a characterization of infinitely divisible distributions with Gaussian component

TL;DR

This paper characterizes when symmetric infinitely divisible distributions contain a Gaussian component, providing conditions useful for approximating sums of random variables and highlighting cases where stable distributions outperform Gaussian approximations.

Contribution

It offers a necessary and sufficient condition for symmetric infinitely divisible distributions to have a Gaussian component, advancing understanding of distribution approximation.

Findings

Provides a criterion for Gaussian component presence in symmetric infinitely divisible distributions.

Shows stable distributions can be better approximations than Gaussian for certain finite variance sums.

Highlights applications in approximating sums of random variables.

Abstract

We give a necessary and sufficient condition for symmetric infinitely divisible distribution to have Gaussian component. The result can be applied to approximation the distribution of finite sums of random variables. Particularly, it shows that for a large class of distributions with finite variance stable approximation appears to be better than Gaussian. keywords: infinitely divisible distributions; Gaussian component; approximations of sums of random variables.