Scaling limit theorem for mixed free and Boolean convolution powers

We prove a scaling limit theorem for a double sequence of probability measures involving additive free convolution $\boxplus$ and additive Boolean convolution $\uplus$.

Let $\mu$ be a probability measure on $\mathbb{R}$ with mean zero and variance one, and let $M=M(N)>0$ satisfy $MN^{\alpha+1/2}\to t>0$.

We study the weak limits, as $N\to \infty$, of the double arrays $D_{N^\alpha}((\mu^{\boxplus N})^{\uplus M})$.

We show that the limit distribution is the Cauchy distribution with scale parameter $t$ if $\alpha>-1/2$, the $t$-fold Boolean convolution power of the standard semicircle law if $\alpha=-1/2$, and the point mass at the origin if $\alpha<-1/2$.