On spectral representation of multivariate stable processes

Let $X(t)$, $t\in\mathbf{R}$, be a symmetric $\alpha$-stable process with independent increments, taking values in $\mathbf{R}^n$. Let $\mathcal{A}=\overline{\operatorname{sp}}\{X(t)-X(s),\,t,s\in\mathbf{R}\}$. Each $Y\in\mathcal{A}$ is a stable vector, and
$$ \mathbf{E}\exp(i\gamma\cdot Y)=\exp\left(-\int_S |\langle\gamma,s\rangle|^\alpha\,d\Gamma_Y(s)\right), $$
where $S$ is a unit sphere in $\mathbf{R}^n$. In this work we prove that there is a unique bimeasure $\pi(\cdot,\cdot)$ on $\mathcal{B}(\mathbf{R})\times\mathcal{B}(S)$ such that for each $Y\in\mathcal{A}$ there is a function $g\in L^\alpha(\pi(\cdot,\mathbf{R}^n))$ such that
$$ \Gamma_Y(\cdot)=\int|g(t)|^\alpha\pi(dt,\cdot). $$
Some applications of this representation are also discussed.