Stochastic comparisons of Tracy-Widom $\beta$ distributions

TL;DR

This paper establishes a stochastic comparison and ordering of Tracy-Widom distributions across different beta parameters, revealing how the distribution's behavior changes as beta increases.

Contribution

It introduces a stochastic comparison framework for Tracy-Widom distributions with varying beta, highlighting how larger beta values lead to smaller distributions after scaling.

Findings

Tracy-Widom distributions decrease as beta increases

Distribution orderings are established across different beta values

Scaling factors are identified for distribution comparisons

Abstract

We give a stochastic comparison and ordering of the Tracy-Widom distribution with parameter $\beta$. In particular, we show that as $\beta$ grows, the Tracy-Widom random variables get smaller modulo a multiplicative coefficient.