\sum_{k = 0} ^{n} \sum_{\substack{j_0 + j_1 + \cdots +j_k = n - k, \\  0 \leq j_l, ~0 \leq l \leq k }} R_1 ^k R_2 ^{n-k} \\
           &= (R_1 + R_2)^n,  
\end{split}$$ since $\sum_{\substack{j_0 + j_1 + \cdots +j_k = n - k, \\  0 \leq j_l, ~0 \leq l \leq k }} = \frac{n!}{k!(n-k)!}$. We replace $\rho$ by $\mu$ and $\sigma$ by $\mu^{\rhd N}$, and then the above inequality becomes $$|m_n(\mu^{\rhd N+1})| \leq (N+1)^n R^n.$$

Let $\mu$ be a probability measure in $\mathcal{P}_1$ with $a = {\text{\normalfont Re~}}\!\big( m_1(\mu)\big)$ and $b = {\text{\normalfont Im~}}\!\big( m_1 (\mu)\big)$. Then $\mu_N^M := (D_\frac{1}{N}\mu)^{\rhd N}$ converges weakly to the Cauchy distribution $\mu_{a,b}$. If $b=0$ we understand that $\mu_{a,0} = \delta_a$.

Applying \[prop3\], we get $K_1^M(\mu_N^M) = a + bi$ and $K_n^M(\mu_N^M)=N^{-(n-1)}K_n^M(\mu) \to 0$ as $N \to \infty$ for $n \geq 2$. The limit complex cumulants are the complex monotone cumulants of the Cauchy distribution $\mu_{a,b}$, and therefore $m_n(\mu_N^M)$ converges to $(a+bi)^n$ as $N \to \infty$. Therefore the generating function $G_{\mu_N^M}(z) = \sum_{n=0} ^\infty\frac{m_n(\mu_N^M)}{z^{n+1}}$ converges to $G_{\mu_{a,b}}(z) =\sum_{n=0} ^\infty\frac{(a+bi)^n}{z^{n+1}}$ in the sense of formal power series. Moreover, the estimation $$\Big|\frac{m_n(\mu_N^M)}{z^{n+1}}\Big| \leq \frac{R^n}{|z|^{n+1}}$$ holds for some $R > 0$ from Lemma \[lem1\] and 