On spectral representation of multivariate stable processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 39 (1994) no. 3, pp. 605-617
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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.
Keywords:
multivariate stable process, independent increments, spectral representation, bimeasure, spectral measure, symmetric measure.
@article{TVP_1994_39_3_a8,
author = {A. Soltani},
title = {On spectral representation of multivariate stable processes},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {605--617},
publisher = {mathdoc},
volume = {39},
number = {3},
year = {1994},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TVP_1994_39_3_a8/}
}
A. Soltani. On spectral representation of multivariate stable processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 39 (1994) no. 3, pp. 605-617. http://geodesic.mathdoc.fr/item/TVP_1994_39_3_a8/