A phase transition for tails of the free multiplicative convolution powers

TL;DR

This paper investigates the tail behavior of measures under free multiplicative convolution powers, revealing a phase transition at tail index 1 and characterizing tails via the S-transform and Lévy measures.

Contribution

It introduces a phase transition in tail behavior for free multiplicative convolution powers and characterizes tail decay using the S-transform and Lévy measure analysis.

Findings

Identifies a phase transition at tail index 1 for free multiplicative convolution powers.

Provides a description of tail behavior in terms of the S-transform at zero.

Establishes a free analog of Breiman's lemma for tail analysis.

Abstract

We study the behavior of the tail of a measure $\mu^{\boxtimes t}$, where $\boxtimes t$ is the $t$-fold free multiplicative convolution power for $t\geq 1$. We focus on the case where $\mu$ is a probability measure on the positive half-line with a regularly varying tail i.e. of the form $x^{-\alpha} L(x)$, where $L$ is slowly varying. We obtain a phase transition in the behavior of the tail of $\mu^{\boxplus t}$ between regimes $\alpha<1$ and $\alpha>1$. Our main tool is a description of the regularly varying tails of $\mu$ in terms of the behavior of the corresponding $S$-transform at $0^-$. We also describe the tails of $\boxtimes$ infinitely divisible measures in terms of the tails of corresponding L\'evy measure, treat symmetric measures with regularly varying tails and prove the free analog of the Breiman lemma.