On tails of symmetric and totally asymmetric $\alpha$-stable distributions

TL;DR

This paper analyzes the tail behavior of symmetric and totally asymmetric alpha-stable distributions, providing precise estimates and identifying the transition point from Gaussian to Pareto-like tails as alpha approaches 2.

Contribution

It offers explicit tail estimates for alpha-stable distributions and pinpoints the exact transition from Gaussian to Pareto tails near alpha equals 2.

Findings

Tail estimates are given up to universal constants.

The transition point from Gaussian to Pareto tail behavior is precisely identified.

Results are applicable for alpha close to 2.

Abstract

We estimate up to universal constants tails of symmetric and totally asymmetric 1-dimensional $\alpha$-stable distributions in terms of functions of the parameters of these distributions. In particular, for values of $\alpha$ close to $2$ we specify where exactly the tail changes from being Gaussian and starts to behave like in the Pareto distribution