\operatorname*{\mathbb{R}}$. Therefore $\mu_N^T$ converges to $\mu_{a,b}$ weakly.

We use the characterization (\[boxplus\]) to prove the weak convergence of $\mu_N^F$. It is easy to prove that $$F_\mu(z)= z -m_1(\mu) +f(z),~~ f(z) \to 0 \text{~as~} z \to \infty  \text{~non tangentially satisfying~} z \in {\mathbb{C}_{-}},$$ by using the Nevanlinna representation $F_\nu(z) = z - m_1(\nu) + \int_{\operatorname*{\mathbb{R}}}\frac{1}{x-z}\tau(dx)$ for $\nu$, where $\tau$ is a positive finite measure [@Akh]. The right inverse can be written as $F_\mu^{-1}(z) = z +m_1(\mu) + g(z)$. We show that $g(z) \to 0$ in the non tangential limit $z \to \infty$. The equality $F_\mu (F_\mu^{-1}(z)) = z$ for $z \in \Omega$ becomes $$0 = g(z) + f(z + m_1(\mu) + g(z)).$$ Since $\frac{F_\mu^{-1}(z)}{z} \to 1$ in the non tangential limit $z \to \infty$ (Corollary 5.5 in [@Be-Vo]), we get $\frac{g(z)}{z} \to 0$. Then $ f(z + m_1(\mu) + g(z)) = f(z (1 + \frac{m_1(\mu)}{z} + \frac{g(z)}{z}) ) \to 0$. Therefore, $g(z) \to 0$ in the non tangential limit. Finally we obtain $$\begin{split}
F^{-1}_{\mu_N^F}(z) &= F^{-1}(Nz) - (N-1)z \\
                    &=  z + m_1 (\mu) + g(Nz) \\
                &\to z + m_1(\mu)   
                    \end{split}$$ as $N \to \infty$ for all $z \in {\mathbb{C}_{-}}$. Thus $\mu_N^F$ converges to $\mu_{a, b}$ from Proposition 5.7 in 