free (additive) convolution of $\mu$ and $\nu$. Thus, $\mu\boxplus\nu$ is the distribution of $X+Y$, where $X$ and $Y$ are free random variables with distributions $\mu$ and $\nu$, respectively. Denote by $\phi_\mu$ the Voiculescu transform of $\mu$ which satisfies the identity $\phi_{\mu\boxplus\nu}=\phi_\mu+\phi_\nu$ in some truncated cone in the upper half-plane $\mathbb{C}^+$.

For $n\in\mathbb{N}$, the $n$-fold free convolution $\mu\boxplus\cdots\boxplus\mu$ is denoted by $\mu^{\boxplus n}$. It was shown in \[\[NS\]\] that the discrete semigroup $\{\mu^{\boxplus
n}:n\in\mathbb{N}\}$ can be embedded in a continuous family $\{\mu^{\boxplus p}:p\geq1\}$ which satisfies $\mu^{\boxplus
p_1}\boxplus\mu^{\boxplus p_2}=\mu^{\boxplus(p_1+p_2)}$, $p_1,p_2\geq1$. Any measure in this family satisfies $\phi_{\mu^{\boxplus p}}=p\phi_\mu$ in some truncated cone in $\mathbb{C}^+$. We refer the reader to \[\[BB1\],\[BB2\], and \[HV3\]\] for complete developments on the existence of this continuous family. In the full generalization, Belinschi and Bercovici used the subordination function to construct the measure $\mu^{\boxplus p}$, $p>1$, and obtained certain regularity properties. In \[\[Huang\]\], an explicit formula for the density of $(\mu^{\boxplus p})^{\mathrm{ac}}$ was provided and the relation between the supports of $\mu$ and $\mu^{\boxplus p}$ was analyzed. As a consequence, the number $n(p)$ of components in the support of $\mu^{\boxplus p}$ was shown to be a decreasing function of $p$.

An important class of measures in $\mathcal{M}$ is 