moments in two different ways: one by means of analytic continuation of the Fourier transforms from $z > 0$ to $\operatorname*{\mathbb{C}}$, and the other by analytic continuation of Stieltjes transforms from ${\mathbb{C}_{-}}$ to ${\mathbb{C}_{-}}\cup \{z \in \operatorname*{\mathbb{C}}; |z|>R \}$ for some $R>0$. While these transforms have different analytic properties, such defined moments turn out to coincide. Then four kinds of cumulants can be defined and satisfy the moment-cumulant formulae. These moments and cumulants are complex numbers and we call them *complex moments* and *complex cumulants* respectively. Then the Cauchy distributions turn out to have the same complex cumulants in any sense of tensor, free, Boolean and monotone. This reveals a universal role of Cauchy distributions. In Section \[Lim\] we prove that the sum of i.i.d. random variables divided by the number of them converges to a Cauchy distribution, using complex moments and cumulants. This holds in any case of tensor, free, Boolean and monotone convolutions. In particular the monotone case is quite new since a complex analytic method is not known to examine limit behavior of monotone convolutions of probability measures with infinite moments. In Section \[Cha\] we characterize $\mathcal{P}_1$ in terms of Fourier transforms and Stieltjes transforms; the characterization 