A formula for hidden regular variation behavior for symmetric stable distributions

TL;DR

This paper introduces a formula to describe the decay rates of symmetric stable distributions, revealing hidden regular variation phenomena and illustrating diverse behaviors across different directions.

Contribution

It provides a novel formula linking decay rates to spectral measure support, enhancing understanding of hidden regular variation in symmetric stable distributions.

Findings

Different decay rates observed in various directions

Examples showing sets with non-power-law decay

Illustration of hidden regular variation phenomenon

Abstract

We develop a formula for the power-law decay of various sets for symmetric stable random vectors in terms of how many vectors from the support of the corresponding spectral measure are needed to enter the set. One sees different decay rates in "different directions", illustrating the phenomenon of hidden regular variation. We give several examples and obtain quite varied behavior, including sets which do not have exact power-law decay.