k=0}^{\infty} \frac {(-y)^k D_{\mathbf{v}}^k}{k!} \right) h({\mathbf{x}})= (1-yD_{{\mathbf{v}}})h({\mathbf{x}}),$$ from which the lemma follows.

Note that $({\mathbf{v}}_1,\ldots,{\mathbf{v}}_m) \mapsto h[{\mathbf{v}}_1,\ldots,{\mathbf{v}}_m]$ is affine linear in each coordinate, i.e., for all $p \in {\mathbb{R}}$ and $1\leq i \leq m$: $$\begin{aligned}
& h[{\mathbf{v}}_1,\ldots,(1-p){\mathbf{v}}_i+p{\mathbf{v}}_i',\ldots, {\mathbf{v}}_m] \\
= &(1-p)h[{\mathbf{v}}_1,\ldots,{\mathbf{v}}_i,\ldots, {\mathbf{v}}_m] +ph[{\mathbf{v}}_1,\ldots,{\mathbf{v}}_i',\ldots, {\mathbf{v}}_m].\end{aligned}$$ Hence if ${\mathsf{X}}_1, \ldots, {\mathsf{X}}_m$ are independent random variables in ${\mathbb{R}}^n$, then $$\label{mixedexp}
{\mathbb{E}}h[{\mathsf{X}}_1,\ldots,{\mathsf{X}}_m] = h[{\mathbb{E}}{\mathsf{X}}_1,\ldots,{\mathbb{E}}{\mathsf{X}}_m].$$

\[mixedchar\] Let $h({\mathbf{x}})$ be hyperbolic with respect to ${\mathbf{e}}\in {\mathbb{R}}^n$, let $V_1, \ldots, V_m$ be finite sets of vectors in $\Lambda_+$, and let ${\mathbf{w}}\in {\mathbb{R}}^{n+m}$. For ${\mathbf{V}}=({\mathbf{v}}_1,\ldots, {\mathbf{v}}_m) \in  V_1\times \cdots \times V_m$, let $$f({\mathbf{V}};t) := h[{\mathbf{v}}_1,\ldots, {\mathbf{v}}_m](t{\mathbf{e}}+{\mathbf{w}}).$$ Then $\{f({\mathbf{V}};t)\}_{{\mathbf{V}}\in V_1\times \cdots \times V_m}$ is a compatible family.

In particular if in addition all vectors in $V_1 \cup \cdots \cup V_m$ have rank at most one, and $$g({\mathbf{V}};t) := h(t{\mathbf{e}}+{\mathbf{w}}- \alpha_1{\mathbf{v}}_1-\cdots-\alpha_m{\mathbf{v}}_m),$$ where ${\mathbf{w}}\in {\mathbb{R}}^n$ and $(\alpha_1,\ldots, \alpha_m)\in {\mathbb{R}}^m$, then $\{g({\mathbf{V}};t)\}_{{\mathbf{V}}\in V_1\times \cdots \times V_m}$ is a compatible family.

Let ${\mathsf{X}}_1 \in V_1, \ldots, {\mathsf{X}}_m \in V_m$ be independent random variables. Then the polynomial ${\mathbb{E}}h[{\mathsf{X}}_1, \ldots, {\mathsf{X}}_m]= h[{\mathbb{E}}{\mathsf{X}}_1,\ldots,{\mathbb{E}}{\mathsf{X}}_m]$ is hyperbolic with respect to $({\mathbf{e}}, 0,\ldots,0)$ b