if $t\geq0$. In addition, if $0<t\leq\mathrm{Ind}(\mu)$ then the measure $\mu^{\uplus t}$ is $\boxplus$-infinitely divisible and its Voiculescu transform can be expressed as $$\label{N2}
\phi_{\mu^{\uplus t}}=E_{\mu_t^{\uplus t}}=
E_{(\mu^{\uplus(2t)})^{\boxplus1/2}}.$$

It is clear that the measure $\mu_t$ in (\[N1\]) is defined and (\[N\]) holds if $-1<t\leq0$. If $t=\mathrm{Ind}(\mu)<\infty$ then $1-\mathrm{Ind}(\mu^{\uplus(t+1)})=1/(t+1)$ by Lemma \[3.4\], whence the measure $\mu_t$ in (\[N1\]) is defined by Proposition \[3.5\]. The same result also holds for $0<t<\mathrm{Ind}(\mu)\in(0,\infty]$, and hence the identity $\mu=\mathbb{B}_t(\mu_t)$ holds. Next, observe that $\mathrm{Ind}(\mu^{\uplus t})\geq1$ for $0<t\leq\mathrm{Ind}(\mu)$, and therefore $\mu^{\uplus t}$ is $\boxplus$-infinitely divisible. The first equality in (\[N2\]) follows by replacing $\mu$ with $\mu^{\uplus(t+1)}$ and letting $p=1/(t+1)$ in Proposition \[3.5\], while the second equality follows from (\[bijection\]).

Next, we relate the limit laws to $\boxplus$-divisibility indicators.

\[limit\] Let $q>0$ and $\{\mu_n\}$ be a sequence of measures in $\mathcal{M}$ such that $\mu_n\to\mu$ weakly as $n\to\infty$ for some $\mu\in\mathcal{M}$. Then the following statements hold.

1.  [The inequality $\lim\sup_n\mathrm{Ind}(\mu_n)\leq\mathrm{Ind}(\mu)$ holds.]{}

2.  [For any $p>0$ with $1-\inf_n\mathrm{Ind}(\mu_n)\leq p$, $\mu_n^{\boxplus
    p}\to\mu^{\boxplus p}$ weakly as $n\to\infty$.]{}

3.  [The measures $\mu_n^{\uplus q}\to\mu^{\uplus q}$ weakly as $n\to\infty$.]{}

The measure $\mu_n^{\boxplus p}$ in (2) is defined for all $n$ by Proposition \[3.5\], whence (2) holds by \[\[HV2\], Proposition 5.7\]. The assertion (3) holds by 