by the coefficients of $N$ in $m_n(\mu^{\ast N})$, $m_n(\mu^{\boxplus N})$, $m_n(\mu^{\uplus N})$ and $m_n(\mu^{\rhd N})$. If there is no confusion, we call them cumulants for simplicity.

We find that the four kinds of cumulants of the Cauchy distribution $\mu_{a,b}$ are given by $$K_n^T(\mu_{a,b}) = K_n^F(\mu_{a,b}) = K_n^B(\mu_{a,b})=K_n^M(\mu_{a,b})=\begin{cases} &a+bi,~~n=1, \\ & 0,~~n \geq 2. \end{cases}$$ This helps us to understand a universal role of Cauchy distributions in noncommutative probability theory.

From now on we show basic properties of complex moments and cumulants.

\[prop11\] ${\text{\normalfont Im~}}\!(m_1(\mu)) \geq 0$ for all $\mu \in \mathcal{P}_2$.

Let $\mu = \nu + \lambda$ be a decomposition in (\[decomp\]). We have only to consider $\lambda$. By Theorem \[thm1\] $m_1(\lambda)$ is calculated as $$\begin{split}
m_1(\lambda) &= \int_{\gamma_R} z p(z)dz \\
              &= \sum_{n = 2}^\infty a_n \int_{\gamma_R} z^{1-n}dz \\
              &= i \sum_{n = 2}^\infty a_n \int_{0}^\pi R^{2-n}e^{i(2-n)\theta} d\theta \\ 
  &= i\pi a_2 - \sum_{n = 3}^\infty a_n  R^{2-n}\frac{1 - (-1)^n}{2-n}.
\end{split}$$ Since $p(x)$ is a positive function, $a_2 \geq 0$. Therefore, ${\text{\normalfont Im~}}\!(m_1(\mu)) = \pi a_2 \geq 0$.

The above fact will be used in Section \[Lim\] to formulate convergence of probability measures to Cauchy distributions.

The following property follows from the integral representation of complex moments in Theorem 