$E_{\mu^{\uplus n}}=nE_\mu$. This can be extended naturally to the case when the exponent $n$ is not an integer. That is, for every $q\geq0$ the $q$-th $\uplus$-convolution power $\mu^{\uplus q}$ is defined as the unique measure in $\mathcal{M}$ satisfying $$E_{\mu^{\uplus q}}=qE_\mu.$$

The following theorem builds the connection between $\boxplus$-infinitely divisible measures and the Boolean convolution, which was thoroughly investigated in \[\[BP\]\].

\[thm2.2\] Let $\{\mu_n\}$ be a sequence in $\mathcal{M}$ and let $k_1<k_2<\cdots$ be a sequence of positive integers. Then the following statements $(1)$-$(3)$ are equivalent:

1.  [$\mu_n^{\boxplus k_n}\to\mu_{\boxplus}^{c,\rho}$ weakly as $n\to\infty$;]{}

2.  [$\mu_n^{\uplus k_n}\to\mu_{\uplus}^{c,\rho}$ weakly as $n\to\infty$;]{}

3.  [the measures $$k_n\frac{s^2}{s^2+1}\;d\mu_n(s)\to d\rho(s)$$ weakly as $n\to\infty$ and $$\lim_{n\to\infty}\int_\mathbb{R}\frac{k_ns}{s^2+1}\;d\mu_n(s)=c.$$]{}

If $(1)$-$(3)$ hold then $\mu_{\boxplus}^{c,\rho}$ is $\boxplus$-infinitely divisible and $$\phi_{\mu_{\boxplus}^{c,\rho}}(z)=E_{\mu_{\uplus}^{c,\rho}}(z)=c+\int_\mathbb{R}\frac{1+sz}{z-s}\;d\rho(s),
\;\;\;\;\;z\in\mathbb{C}^+.$$

For $\mu\in\mathcal{M}$, Theorem \[thm2.2\] shows that $(\mu^{\uplus1/n})^{\boxplus n}$ converges weakly to some $\boxplus$-infinitely divisible measure $\mathbb{B}(\mu)$ satisfying $\phi_{\mathbb{B}(\mu)}=E_\mu$. Conversely, for any $\boxplus$-infinitely divisible measure $\nu$ the sequence $(\nu^{\boxplus1/n})^{\uplus n}$ converges weakly to some $\mu$ satisfying $\phi_\nu=E_\mu$. Since $E$ determines the measure uniquely, the map $\mathbb{B}$ induces a bijective map from $\mathcal{M}$ onto the set of $\boxplus$-infinitely divisible measures. This map $\mathbb{B}$ is called the Boolean Bercovici-Pata bijection, which coincides with $\mathbb{B}_1$ as indicated in the introduction.

In the study of $\boxplus$-infinitely divisible measures, 