Suppose that I have an $n\times 1$ random vector $X=(X_1,X_2,\ldots,X_n)'$. For $\xi=(\xi_1,\ldots,\xi_n)'\in\mathbb{R}^n$, we can define the familiar generating functions $$ M_X(\xi)=E\Big[\exp\Big(\sum_i\xi_iX_i\Big)\Big],\quad K_X(\xi)=\log[M_X(\xi)]. $$ Then the cross cumulants can be read off the Taylor expansion of $K_X(\xi)$: $$ K_X(\xi)=\sum_i\xi_i\kappa^i+\sum_{i,j}\xi_i\xi_j\frac{\kappa^{i,j}}{2!}+\sum_{i,j,k}\xi_i\xi_j\xi_k\frac{\kappa^{i,j,k}}{3!}+\cdots \tag{$*$} $$ Here the cross cumulants are the $\kappa$'s terms.
Question: suppose that I'm interested in $\kappa^{1,1}$ from ($*$). Will it be the same as the $\kappa^{1,1}$ term from a similar Taylor expansion for $K_Z(\cdot)$ where $Z=(X_1,\ldots,X_m)'$, $m<n$, is a strict subvector of $X$? If "YES", can you provide a proof or point me to one? Intuition is also appreciated.