$\mathcal{P}_1$ is closed under the four kinds of convolutions, and hence the above cumulants such as $K_n^T(\mu_1 \ast \mu_2)$ are well-defined.

This proof is also the same as the usual one. See [@H-S; @H-S2; @Shi; @Spe2; @S-W] for details.

Convergence of probability measures to Cauchy distribution {#Lim}
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Now we prove that the distribution of $\frac{X_1 + \cdots + X_N}{N}$ converges weakly to Cauchy distributions for i.i.d. random variables $X_i$ with the distribution in $\mathcal{P}_1$ or $\mathcal{P}_2$. We consider only probability measures since difficulty arises to formulate the results in terms of unbounded operators. It is known that $\mathcal{P}_2$ belong to the domain of attractor of Cauchy distributions in tensor, free and Boolean probability theories [@Be-Pa; @K-G]. There are however two merits in the approach of this paper: the proof is simple; the two parameters $a$ and $b$ appearing in the limit distributions can be explicitly calculated in terms of complex moments.

Before proving them, we introduce a tool in free probability. A domain of $F^{-1}_\mu$ for a probability measure $\mu$ is defined as follows. It is known that there exists a domain $\Omega$ such that $F_\mu$ has the right inverse $F_\mu^{-1}$ on $\Omega$. It is possible to choose $\Omega$ 