Consider a $d$-dimensional random vector $\mathbf X$ with laplace transfrom $$L(\mathbf t) = \left(1 + \langle \mathbf \theta, \mathbf t \rangle\right)^{-\alpha}$$
Where $\mathbf \theta$ is a $d$-dimensional vector of non-negatives reals (some can be zero, not all), and $\alpha$ is a strictly positive real (that cannot be zero). The coresponding random vector writes as $\mathbf X = \left(\theta_1 Y,...,\theta_d Y\right)$ for $Y$ a random variable following a gamma distribution with shape $\alpha$ and unit scale, that is with laplace tranform $\left(1 + t\right)^{-\alpha}$. Therefore, $\mathbf X$ is a comonotonous vector, living on a straigth half-line in $\mathbb R^d$.
Question: Considering $L$ as a function from $\mathbb C^d$ to $\mathbb C$, what is the region of convergence of $L$ ? Where are it's singularities ?
$\alpha >0$ and $$\int_{\Bbb{R}^n} e^{-\langle t, x\rangle} d\mu_X(x)= \int_{ \Bbb{R}}e^{-\langle t, \theta y\rangle} d\mu_Y(y)= \int_{[0,\infty)}e^{-\langle t, \theta \rangle y} \frac{y^{\alpha-1} e^{-y}}{\Gamma(\alpha)} dy $$ converges for $\Re(\langle \theta,t\rangle) > -1$ and diverges for $\Re(\langle \theta,t\rangle) \le -1$.
It extends analytically to $\{ t\in \Bbb{C}^n,\langle \theta,t\rangle\in U\}$ where $U$ is any simply connected open of $\Bbb{C}-\{-1\}$.
If $\alpha$ is an integer then it extends to a meromorphic function on $\Bbb{C}^n$ with only one pole of order $\alpha$ at the hypersurface $\langle \theta,t\rangle=-1$.