Scaling limit theorem for mixed free and Boolean convolution powers

Abstract

We prove a scaling limit theorem for a double sequence of probability measures involving additive free convolution $⊞$ and additive Boolean convolution $\uplus$. Let $\hat{I}¼$ be a probability measure on $\mathbb{R}$ with mean zero and variance one, and let $M=M(N)>0$ satisfy $M{N}^{\hat{I}\pm +1/2}\to t>0$. We study the weak limits, as $N\to \infty$, of the double arrays $D_{N^{\hat{I}}\pm }((\hat{I}{¼}^{⊞N}{)}^{\uplus M})$. We show that the limit distribution is the Cauchy distribution with scale parameter $t$ if $\hat{I}\pm >-1/2$, the $t$-fold Boolean convolution power of the standard semicircle law if $\hat{I}\pm =-1/2$, and the point mass at the origin if $\hat{I}\pm <-1/2$.

9 pages