\}_{n \geq 1} \subset \operatorname*{\mathbb{C}}$, we define $r_\pi:= r_{|V_1|} \cdots r_{|V_k|}$.

Since we defined complex moments via analytic continuations, the following properties hold.

The moment-cumulant formulae hold: $$\begin{aligned}
&m_n (\mu) = \sum_{\pi \in {\mathcal{P}(n)}} K_\pi^T (\mu), \label{cmc} \\
&m_n(\mu) = \sum_{\pi \in {\mathcal{NC}(n)}} K_\pi^F(\mu), \label{fmc} \\
&m_n (\mu) = \sum_{\pi \in {\mathcal{I}(n)}} K_\pi^B(\mu), \label{bmc} \\
&m_n (\mu)= \sum_{(\pi, \lambda) \in {\mathcal{M}(n)}} \frac{K_\pi^M(\mu)}{|\pi|!}. \label{mmc}   \end{aligned}$$

The proof is the same as the usual case where $\mu$ has a compact support; the reader is referred to [@H-S; @Shi; @Spe2; @S-W] and also [@H-S2] for a unified treatment of (\[cmc\]), (\[fmc\]) and (\[bmc\]).

Let $D_c$ be the dilation operator defined by $$\int_{\operatorname*{\mathbb{R}}} f(x) (D_c \mu)(dx) = \int_{\operatorname*{\mathbb{R}}}f(c x) \mu(dx)$$ for all bounded continuous function $f$.

\[prop3\] Let $\mu, ~\mu_1, ~\mu_2 \in \mathcal{P}_1$. The complex cumulants are additive and homogeneous:\
(1) $K_n^T(\mu_1 \ast \mu_2) = K_n^T(\mu_1) + K_n^T(\mu_2)$,   $K_n^T (D_c \mu) = c^n K_n^T(\mu)$,\
(2) $K_n^F(\mu_1 \boxplus \mu_2) = K_n^F(\mu_1) + K_n^F(\mu_2)$,   $K_n^F (D_c \mu) = c^n K_n^F(\mu)$,\
(3) $K_n^B(\mu_1 \uplus \mu_2) = K_n^B(\mu_1) + K_n^B(\mu_2)$,   $K_n^B (D_c \mu) = c^n K_n^B(\mu)$,\
(4) $K_n^M(\mu ^{\rhd N}) = NK_n^M(\mu)$ for any $N \in \operatorname*{\mathbb{N}}$,   $K_n^M (D_c \mu) = c^n K_n^M(\mu)$.

As proved in Corollary \[cor1\], 