A phase transition for tails of the free multiplicative convolution powers

Abstract

We study the behavior of the tail of a measure μ⊠t, where ⊠t is the t-fold free multiplicative convolution power for t≥1. We focus on the case where μ is a probability measure on the positive half-line with a regularly varying right tail i.e. of the form x−αL(x), where L is slowly varying. We obtain a phase transition in the behavior of the right tail of μ⊞t between regimes α<1 and α>1. Our main tool is a description of regular variation of the tail of μ in terms of the behavior of the corresponding S-transform at 0−. We also describe the tails of ⊠ infinitely divisible measures in terms of the tails of corresponding Lévy measure, treat symmetric measures with regularly varying tails and prove the free analog of the Breiman lemma.